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Full text of "Mysticism and logic and other essays"

THIS BOOK IS FROM THE 

LawUr Library 

154 LAKE PROMENADE 
V TORONTO 14 



MYSTICISM AND LOGIC 
AND OTHER ESSAYS 



BY BERTRAND RUSSELL 

The ABC of Relativity 

The Analysis of Matter 

Human Society in Ethics and Politics 

The Impact of Science on Society 

New Hopes for a Changing World 

Authority and the Individual 

Hitman Knowledge 

History of Western Philosophy 

The Principles of Mathematics 

Introduction to Mathematical Philosophy 

The A nalysis of Mind 

Our Knowledge of the External World 

An Outline of Philosophy 

The Philosophy of Leibniz 

An Inquiry into Meaning and Truth 

Logic and Knowledge 

The Problems of Philosophy 

Principia Mathematica 

Common Sense and Nuclear Warfare 
Why I am Not a Christian 

Portraits from Memory 

My Philosophical Development 

Unpopular Essays 

Power 

In Praise of Idleness 
The Conquest of Happiness 

Sceptical Essays 

The Scientific Outlook 

Marriage and Morals 

Education and the Social Order 

On Education 

Freedom and Organization 
Principles of Social Reconstruction 

Roads to Freedom 
Practice and Theory of Bolshevism 

Satan in The Suburbs 
Nightmares of Eminent Persons 



BERTRAND RUSSELL 

MYSTICISM AND 
LOGIC 

AND OTHER ESSAYS 



LONDON 
GEORGE ALLEN & UNWIN LTD 

RUSKIN HOUSE MUSEUM STREET 



First published as "Philosophical Essays" 
October 1910 

Second Edition as "Mysticism and Logic" 

December 1917 

Third Impression - - - - April 1918 

Fourth Impression - - February 1919 
Fifth Impression - - October 1921 

Sixth Impression - - -August 1925 

Seventh Impression- - January 1932 
Eighth Impression ------ 1949 

Ninth Impression - - - - - - 1950 

Tenth Impression - - - - - - 1951 

Eleventh Impression - - 1959 



This book is copyright under the Berne Con 
vention. Apart from any fair dealing for the 
purpose of private study, research, criticism or 
review, as permitted under the Copyright Act, 
1956, no portion may be reproduced by any 
process without written permission. Enquiry 
should be made to the publisher. 




PRINTED IN GREAT BRITAIN 

by Taylor Garnett Evans & Co. Ltd., 
Watford, Herts. 



PREFACE 

r I ""HE following essays have been written and pub- 

1 lished at various times, and my thanks are due to 

the previous publishers for the permission to reprint 



The essay on " Mysticism and Logic " appeared in the 
Hibbert Journal for July, 1914. " The Place of Science 
in a Liberal Education " appeared in two numbers of 
The New Statesman, May 24 and 31, 1913. " The Free 
Man s Worship " and " The Study of Mathematics " 
were included in a former collection (now out of print), 
Philosophical Essays, also published by Messrs. Longmans, 
Green & Co. Both were written in 1902 ; the first appeared 
originally in the Independent Review for 1903, the second 
in the New Quarterly, November, 1907. In theoretical 
Ethics, the position advocated in " The Free Man s 
Worship " is not quite identical with that which I 
hold now : I feel less convinced than I did then of the 
objectivity of good and evil. But the general attitude 
towards life which is suggested in that essay still seems 
to me, in the main, the one which must be adopted in 
times of stress and difficulty by those who have no 
dogmatic religious beliefs, if inward defeat is to be 
avoided. 

The essay on " Mathematics and the Metaphysicians " 
was written in 1901, and appeared in an American maga 
zine, The International Monthly, under the title " Recent 
Work in the Philosophy of Mathematics." Some points 



vt MYSTICISM AND LOGIC 

in this essay require modification in view of later work. 
These are indicated in footnotes. Its tone is partly 
explained by the fact that the editor begged me to make 
the article " as romantic as possible." 

All the above essays are entirely popular, but those 
that follow are somewhat more technical. " On Scientific 
Method in Philosophy " was the Herbert Spencer lecture 
at Oxford in 1914, and was published by the Clarendon 
Press, which has kindly allowed me to include it in this 
collection. " The Ultimate Constituents of Matter " 
was an address to the Manchester Philosophical Society, 
early in 1915, and was published in the Monist in July 
of that year. The essay on " The Relation of Sense-data 
to Physics " was written in January, 1914, and first 
appeared in No. 4 of that year s volume of Scientia, an 
International Review of Scientific Synthesis, edited by 
M. Eugenio Rignano, published monthly by Messrs. 
Williams and Norgate, London, Nicola Zanichelli, 
Bologna, and Felix Alcan, Paris. The essay " On the 
Notion of Cause " was the presidential address to the 
Aristotelian Society in November, 1912, and was pub 
lished in their Proceedings for 1912-13. " Knowledge by 
Acquaintance and Knowledge by Description " was also 
a paper read before the Aristotelian Society, and pub 
lished in their Proceedings for 1910-11. 

LONDON, 



CONTENTS 

Chaptr P*gs 

1. Mysticism and Logic i 

II. The Place of Science in a Liberal Education 33 

III A Free Man s Worship 46 

IV. The Study oj Mathematics 58 

V. Mathematics and the Metaphysicians 74 

VI. On Scientific Method in Philosophy 97 

VII. The Ultimate Constituents of Matter 125 

VIII. The Relation of Sense-data to Physics 145 

IX. On the Motion of Cause 180 

X. Knowledge by Acquaintance and Knowledge 

by Description 209 



MYSTICISM AND LOGIC 
AND OTHER ESSAYS 



MYSTICISM AND LOGIC 

METAPHYSICS, or the attempt to conceive the 
world as a whole by means of thought, has been 
developed, from the first, by the union and conflict of 
two very different human impulses, the one urging men 
towards mysticism, the other urging them towards 
science. Some men have achieved greatness through 
one of these impulses alone, others through the other 
alone : in Hume, for example, the scientific impulse 
reigns quite unchecked, while in Blake a strong hostility 
to science co-exists with profound mystic insight. But 
the greatest men who have been philosophers have felt 
the need both of science arid of mysticism : the attempt 
to harmonise the two was what made their life, and what 
always must, for all its arduous uncertainty, make 
philosophy, to some minds, a greater thing than either 
science or religion. 

Before attempting an explicit characterisation of the 
scientific and the mystical impulses, I will illustrate 
them by examples from two philosophers whose great 
ness lies in the very intimate blending which they 
achieved. The two philosophers I mean are Herac&ius 
and Plato. 



* MYSTICISM AND LOGIC 

Heraclitus, as every one knows, was a believer in 
universal flux : time builds and destroys all things. 
From the few fragments that remain, it is not easy to 
discover how he arrived at his opinions, but there are 
some sayings that strongly suggest scientific observation 
as the source. 

" The things that can be seen, heard, and learned/ he 
says, " are what I prize the most." This is the language 
of the empiricist, to whom observation is the sole guaran 
tee of truth. " The sun is new every day," is another 
fragment ; and this opinion, in spite of its paradoxical 
character, is obviously inspired by scientific reflection, 
and no doubt seemed to him to obviate the difficulty of 
understanding how the sun can work its way under 
ground from west to east during the night. Actual 
observation must also have suggested to him his central 
doctrine, that Fire is the one permanent substance, of 
which all visible things are passing phases. In com 
bustion we see things change utterly, while their flame 
and heat rise up into the air and vanish. 

" This world, which is the same for all," he says, " no 
one of gods or men has made ; but it was ever, is now, 
and ever shall be, an ever-living Fire, with measures 
kindling, and measures going out." 

" The transformations of Fire are, first of all, sea ; and 
half of the sea is earth, half whirlwind." 

This theory, though no longer one which science can 
accept, is nevertheless scientific in spirit. Science, too, 
might have inspired the famous saying to which Plato 
alludes : " You cannot step twice into the same rivers ; 
for fresh waters are ever flowing in upon you." But we 
find also another statement among the extant fragments : 
" We step and do not step into the same rivers ; we are 
and are not." 



MYSTICISM AND LOGIC 3 

The comparison of this statement, which is mystical, 
with the one quoted by Plato, which is scientific, shows 
how intimately the two tendencies are blended in the 
system of Heraclitus. Mysticism is, in essence, little 
more than a certain intensity and depth of feeling in 
regard to what is believed about the universe ; and this 
kind of feeling leads Heraclitus, on the basis e>f his science, 
to strangely poignant sayings concerning life and the 
world, such as : 

" Time is a child playing draughts, the kingly power is 
a child s." 

It is poetic imagination, not science, which presents 
Time as despotic lord of the world, with all the irrespon- 
~ible frivolity of a child. It is mysticism, too, which 
leads IleTaeiitus to assert the identity of opposites : 
" Good and ill are one," he says ; and again : " To God 
all things are fair and good and right, but men hold some 
things wrong and some right." 

Much of mysticism underlies the ethics of Heraclitus. 
It is true that a scientific determinism alone might have 
inspired the statement : " Man s character is his fate " ; 
but only a mystic would have said : 

" Every beast is driven to the pasture with blows " ; 
and again : 

" It is hard to fight with one s heart s desire. What 
ever it wishes to get, it purchases at the cost of soul " ; 
and again : 

" Wisdom is one thing. It is to know the thought by 
which all things are steered through all things." 1 

Examples might be multiplied, but those that have 
been given are enough to show the character of the man : 
the facts of science, as they appeared to him, fed the 

1 All the above quotations are from Burnet s Ecurlv Greek P kilo- 
top hv (2nd ed., 1908), pp. 146-156. 



4 MYSTICISM AND LOGIC 

flame in his soul, and in its light he saw into the depths 
of the world by the reflection of his own dancing swiftly 
penetrating fire. In such a nature we see the true union 
of the mystic and the man of science the highest 
eminence, as I think, that it is possible to achieve in the 
world of thought. 

In Plato, the same twofold impulse exists, though the 
mystic impulse is distinctly the stronger of the two, and 
secures ultimate victory whenever the conflict is sharp. 
His description of the cave is the classical statement of 
belief in a knowledge and reality truer and more real 
than that of the senses : 

" Imagine 1 a number of men living in an underground 
cavernous chamber, with an entrance open to the light, 
extending along the entire length of the cavern, in which 
they have been confined, from their childhood, with their 
legs and necks so shackled that they are obliged to &it 
still and look straight forwards, because their chains 
render it impossible for them to turn their heads round : 
and imagine a bright fire burning some way off, above 
and behind them, and an elevated roadway passing 
between the fire and the prisoners, with a low wall built 
along it, like the screens which conjurors put up in front 
of their audience, and above which they exhibit their 
wonders. 

I have it, he replied. 

Also figure to yourself a number of persons walking 
behind this wall, and carrying with them statues of men, 
and images of other animals, wrought in wood and stone 
and all kinds of materials, together with various other 
articles, which overtop the wall ; and, as you might 
expect, let some of the passers-by be talking, and others 
silent. 

1 Republic, 514, translated by Davies and Vaughan. 



MYSTICISM AND LOGIC * 

You are describing a strange scene, and strange 
prisoners. 

They resemble us, I replied. 

Now consider what would happen if the course of 
nature brought them a release from their fetters, and a 
remedy for their foolishness, in the following manner. 
Let us suppose that one of them has been released, and 
compelled suddenly to stand up, and turn his neck round 
and walk with open eyes towards the light ; and let us 
suppose that he goes through all these actions with pain, 
and that the dazzling splendour renders him incapable of 
discerning those objects of which he used formerly to see 
the shadows. What answer should you expect him to 
make, if some one were to tell him that in those days he 
was watching foolish phantoms, but that now he is some 
what nearer to reality, and is turned towards things more 
real, and sees more correctly ; above all, if he were to 
point out to him the several objects that are passing by, 
and question him, and compel him to answer what they 
are ? Should you not expect him to be puzzled, and to 
regard his old visions as truer than the objects now forced 
upon his notice ? 

Yes, much truer. . . . 

Hence, I suppose, habit will be necessary to enable him 
to perceive objects in that upper world. At first he will 
be most successful in distinguishing shadows ; then he 
will discern the reflections of men and other things in 
water, and afterwards the realities ; and after this he will 
raise his eyes to encounter the light of the moon and stars, 
finding it less difficult to study the heavenly bodies and 
the heaven itself by night, than the sun and the sun s light 
by day. 

Doubtless. 

Last of all, I imagine, he will be able to observe and 



6 MYSTICISM AND LOGIC 

contemplate the nature of the sun, not as it appears in 
water or on alien ground, but as it is in itself in its own 
territory. 

Of course. 

His next step will be to draw the conclusion, that the 
sun is the author of the seasons and the years, and the 
guardian of all things in the visible world, and in a manner 
the cause of all those things which he and his companions 
used to see. 

Obviously, this will be his next step. . . . 

Now this imaginary case, my dear Glancon, you must 
apply in all its parts to our former statements, by com 
paring the region which the eye reveals, to the prison 
house, and the light of the fire therein to the power of the 
sun : and if, by the upward ascent and the contemplation 
of the upper world, you understand the mounting of the 
soul into the intellectual region, you will hit the tendency 
of my own surmises, since you desire to be told what they 
are ; though, indeed, God only knows whether they are 
correct. But, be that as it may, the view which I take of 
the subject is to the following effect. In the world of 
knowledge, the essential Form of Good is the limit of our 
enquiries, and can barely be perceived ; but, when 
perceived, we cannot help concluding that it is in every 
case the source of all that is bright and beautiful, in the 
visible world giving birth to light and its master, and in 
the intellectual world dispensing, immediately and with 
full authority, truth and reason ; and that whosoever 
would act wisely, either in private or in public, must set 
this Form of Good before his eyes." 

But in this passage, as throughout most of Plato s 
teaching, there is an identification of the good with the 
truly real, which became embodied in the philosophical 



MYSTICISM AND LOGIC 7 

tradition, and is still largely operative in our own day. 
In thus allowing a legislative function to the good, Plato 
produced a divorce between philosophy and science, 
from which, in my opinion, both have suffered ever since 
and are still suffering. The man of science, whatever his 
hopes may be, must lay them aside while he studies 
nature ; and the philosopher, if he is to achieve truth 
must do the same. Ethical considerations can only 
legitimately appear when the truth has been ascertained : 
they can and should appear as determining our feeling 
towards the truth, and our manner of ordering our lives 
in view of the truth, but not as themselves dictating what 
the truth is to be. 

There are passages in Plato among those which illus 
trate the scientific side of his mind where he seems 
clearly aware of this. The most noteworthy is the one 
in which Socrates, as a young man, is explaining the 
theory of ideas to Parmenides. 

After Socrates has explained that there is an idea of 
the good, but not of such things as hair and mud and 
dirt, Parmenides advises him " not to despise even tha 
meanest things," and this advice shows the genuine 
scientific temper. It is with this impartial temper that 
the mystic s apparent insight into a higher reality and a 
hidden good has to be combined if philosophy is to realise 
its greatest possibilities. And it is failure in this respect 
that has made so much of idealistic philosophy thin, 
lifeless, and insubstantial. It is only in marriage with 
the world that our ideals can bear fruit : divorced from 
it, they remain barren. But marriage with the world is 
not to be achieved by an ideal which shrinks from fact, 
or demands in advance that the world shall conform to 
its desires. 

Parmenides himself is the source of a peculiarly 



8 MYSTICISM AND LOGIC 

interesting strain of mysticism which pervades Plato s, 
thought the mysticism which may be called " logical " 
because it is embodied in theories on logic. This form of 
mysticism, which appears, so far as the West is con 
cerned, to have originated with Parmenides, dominates 
the reasonings of all the great mystical metaphysicians 
from his day to that of Hegel and his modern disciples. 
Reality, he says, is uncreated, indestructible, unchanging, 
indivisible ; it is " immovable in the bonds of mighty 
chains, without beginning and without end ; since coming 
into being and passing away have been driven afar, and 
true belief has cast them away." The fundamental 
principle of his inquiry is stated in a sentence which 
would not be out of place in Hegel : " Thou canst not 
know what is not that is impossible nor utter it ; for 
it is the same thing that can be thought and that can be." 
And again : "It needs must be that what can be thought 
and spoken of is ; for it is possible for it to be, and it is 
not possible for what is nothing to be." The impossi 
bility of change follows from this principle ; for what is 
past can be spoken of, and therefore, by the principle, 
still is. 

Mystical philosophy, in all ages and in all parts of the 
world, is characterised by certain beliefs which are illus 
trated by the doctrines we have been considering. 

There is, first, the belief in insight as against discur 
sive analytic knowledge : the belief in a way of wisdom, 
sudden, penetrating, coercive, which is contrasted with 
the slow and fallible study of outward appearance by a 
science relying wholly upon the senses. All who are 
capable of absorption in an inward passion must have 
experienced at times the strange feeling of unreality in 
common objects, the loss of contact with daily things, in 
which the solidity of the outer world is lost, and the soul 



MYSTICISM AND LOGIC 9 

seems, in utter loneliness, to bring forth, out of its own 
depths, the mad dance of fantastic phantoms which have 
hitherto appeared as independently real and living. 
This is the negative side of the mystic s initiation : the 
doubt concerning common knowledge, preparing the way 
for the reception of what seems a higher wisdom. Many 
men to whom this negative experience is familiar do not 
pass beyond it, but for the mystic it is merely the gateway 
to an ampler world. 

The mystic insight begins with the sense of a mystery 
unveiled, of a hidden wisdom now suddenly become 
certain beyond the possibility of a doubt. The sense of 
certainty and revelation comes earlier than any definite 
belief. The definite beliefs at which mystics arrive are 
the result of reflection upon the inarticulate experience 
gained in the moment of insight. Often, beliefs which 
have no real connection with this moment become subse 
quently attracted into the central nucleus ; thus in addi 
tion to the convictions which all mystics share, we find, 
in many of them, other convictions of a more local and 
temporary character, which no doubt become amalga 
mated with what was essentially mystical in virtue of 
their subjective certainty. We may ignore such inessential 
accretions, and confine ourselves to the beliefs which all 
mystics share. 

The first and most direct outcome of the moment of 
illumination is belief in the possibility of a way of know 
ledge which may be called revelation or insight or in 
tuition, as contrasted with sense, reason, and analysis, 
which are regarded as blind guides leading to the morass 
of illusion. Closely connected with this belief is the 
conception of a Reality behind the world of appearance 
and utterly different from it. This Reality is regarded 
with an admiration often amounting to worship ; it is 



io MYSTICISM AND LOGIC 

felt to be always and everywhere close at hand, thinly 
veiled by the shows of sense, ready, for the receptive 
mind, to shine in its glory even through the apparent 
folly and wickedness of Man. The poet, the artist, and 
the lover are seekers after that glory : the haunting 
beauty that they pursue is the faint reflection of its sun. 
But the mystic lives in the full light of the vision : what 
others dimly seek he knows, with a knowledge beside 
which all other knowledge is ignorance. 

The second characteristic of mysticism is its belief in 
unity, and its refusal to admit opposition or division 
anywhere. We found Heraclitus saying " good and ill 
are one " ; and again he says, " the way up and the way 
down is one and the same." The same attitude appears 
in the simultaneous assertion of contradictory pro 
positions, such as : " We step and do not step into the 
same rivers ; we are and are not." The assertion of Par- 
menides, that reality is one and indivisible, comes from 
the same impulse towards unity. In Plato, this impulse 
is less prominent, being held in check by his theory of 
ideas ; but it reappears, so far as his logic permits, in the 
doctrine of the primacy of the Good. 

A third mark of almost all mystical metaphysics is the 
denial of the reality of Time. This is an outcome of the 
denial of division ; if all is one, the distinction of past 
and future must be illusory. We have seen this doctrine 
prominent in Parmenides ; and among moderns it is 
fundamental in the systems of Spinoza and Hegel. 

The last of the doctrines of mysticism which we have 
to consider is its belief that all evil is mere appearance, 
an illusion produced by the divisions and oppositions of 
the analytic intellect. Mysticism does not maintain that 
such things as cruelty, for example, are good, but it 
denies that they are real : they belong to that lower 



MYSTICISM AND LOGIC II 

world of phantoms from which we are to be liberated by 
the insight of the vision. Sometimes for example in 
Hegel, and at least verbally in Spinoza not only evil, 
but good also, is regarded as illusory, though nevertheless 
the emotional attitude towards what is held to be Reality 
is such as would naturally be associated with the belief 
that Reality is good. What is, in all cases, ethically 
characteristic of mysticism is absence of indignation or 
protest, acceptance with joy, disbelief in the ultimate 
truth of the division into two hostile camps, the good and 
the bad. This attitude is a direct outcome of the nature 
of the mystical experience : with its sense of unity is 
associated a feeling of infinite peace. Indeed it may be 
suspected that the feeling of peace produces, as feelings 
do in dreams, the whole system of associated beliefs 
which make up the body of mystic doctrine. But this is 
a difficult question, and one on which it cannot be hoped 
that mankind will reach agreement. 

Four questions thus arise in considering the truth or 
falsehood of mysticism, namely : 

I. Are there two ways of knowing, which may be called 
respectively reason and intuition ? And if so, is either to 
be preferred to the other ? 

II. Is all plurality and division illusory ? 

III. Is time unreal ? 

IV. What kind of reality belongs to good and evil ? 

On all four of these questions, while fully developed 
mysticism seems to me mistaken, I yet believe that, by 
sufficient restraint, there is an element of wisdom to be 
learned from the mystical way of feeling, which does not 
seem to be attainable in any other manner. If this is the 
truth, mysticism is to be commended as an attitude 
towards life, not as a creed about the world. The meta- 



12 MYSTICISM AND LOGIC 

physical creed, I shall maintain, is a mistaken outcome 
of the emotion, although this emotion, as colouring and 
informing all other thoughts and feelings, is the inspirer 
of whatever is best in Man. Even the cautious and 
patient investigation of truth by science, which seems 
the very antithesis of the mystic s swift certainty, may 
be fostered and nourished by that very spirit of reverence 
in which mysticism lives and moves. 



I. REASON AND INTUITION 1 

Of the reality or unreality of the mystic s world I know 
nothing. I have no wish to deny it, nor even to declare 
that the insight which reveals it is not a genuine insight. 
What I do wish to maintain and it is here that the 
scientific attitude becomes imperative is that insight, 
untested and unsupported, is an insufficient guarantee of 
truth, in spite of the fact that much of the most important 
truth is first suggested by its means. It is common to 
speak of an opposition between instinct and reason ; in 
the eighteenth century, the opposition was drawn in 
favour of reason, but under the influence of Rousseau and 
the romantic movement instinct was given the preference, 
first by those who rebelled against artificial forms of 
government and thought, and then, as the purely 
rationalistic defence of traditional theology became 
increasingly difficult, by all who felt in science a menace 
to creeds which they associated with a spiritual outlook 
on life and the world. Bergson, under the name of 
"intuition," has raised instinct to the position of sole 

1 This section, and also one or two pages in later sections, have been 
printed in a course of Lowell lectures On our knowledge of the external 
world, published by the Open Court Publishing Company. But I have 
left them here, as this is the context for which they were originally 
written. 



MYSTICISM AND LOGIC 13 

arbiter of metaphysical truth. But in fact the opposi 
tion of instinct and reason is mainly illusory. Instinct, 
intuition, or insight is what first leads to the beliefs 
tfhich subsequent reason confirms or confutes ; but the 
Confirmation, where it is possible, consists, in the last 
analysis, of agreement with other beliefs no less in 
stinctive. Reason is a harmonising, controlling force 
rather than a creative one. Even in the most purely 
logical realm, it is insight that first arrives at what is 
new. 

Where instinct and reason do sometimes conflict is in 
regard to single beliefs, held instinctively, and held with 
such determination that no degree of inconsistency with 
other beliefs leads to their abandonment. Instinct, like 
all human faculties, is liable to error. Those in whom 
reason is weak are often unwilling to admit this as 
regards themselves, though all admit it in regard to 
others. Where instinct is least liable to error is in 
practical matters as to which right judgment is a help to 
survival : friendship and hostility in others, for instance, 
are often felt with extraordinary discrimination through 
very careful disguises. But even in such matters a wrong 
impression may be given by reserve or flattery ; and in 
matters less directly practical, such as philosophy deals 
with, very strong instinctive beliefs are sometimes wholly 
mistaken, as we may come to know through their per 
ceived inconsistency with other equally strong beliefs. 
It is such considerations that necessitate the harmonising 
mediation of reason, which tests our beliefs by their 
mutual compatibility, and examines, in doubtful cases, 
the possible sources of error on the one side and on the 
other. In this there is no opposition to instinct as a 
whole, but only to blind reliance upon some one interest 
ing aspect of instinct to the exclusion of other more 



I 4 MYSTICISM AND LOGIC 

commonplace but not less trustworthy aspects. It is 
such one-sidedness, not instinct itself, that reason aims 
at correcting. 

These more or less trite maxims may be illustrated by 
application to Bergson s advocacy of " intuition " as 
against " intellect." There are, he says, " two profoundly 
different ways of knowing a thing. The first implies that 
we move round the object : the second that we enter 
into it. The first depends on the point of view at which 
we are placed and on the symbols by which we express 
ourselves. The second neither depends on a point of 
view nor relies on any symbol. The first kind of knowledge 
may be said to stop at the relative ; the second, in those 
cases where it is possible, to attain the absolute." 1 The 
second of these, which is intuition, is, he says, " the kind 
of intellectual sympathy by which one places oneself 
within an object in order to coincide with what is unique 
in it and therefore inexpressible " (p. 6). In illustration, 
he mentions self-knowledge : " there is one reality, at 
least, which we all seize from within, by intuition and 
not by simple analysis. It is our own personality in its 
flowing through time our self which endures " (p. 8). 
The rest of Bergson s philosophy consists in reporting, 
through the imperfect medium of words, the knowledge 
gained by intuition, and the consequent complete con 
demnation of all the pretended knowledge derived from 
science and common sense. 

This procedure, since it takes sides in a conflict of 
instinctive beliefs, stands in need of justification by 
proving the greater trustworthiness of the beliefs on one 
side than of those on the other. Bergson attempts this 
justification in two ways, first by explaining that intellect 
is a purely practical faculty to secure biological success, 

1 Introduction to Metaphysics, p. A. 



MYSTICISM AND LOGIC 15 

secondly by mentioning remarkable feats of instinct in 
animals and by pointing out characteristics of the world 
which, though intuition can apprehend them, are 
baffling to intellect as he interprets it. 

Of Bergson s theory that intellect is a purely practical 
faculty, developed in the struggle for survival, and not a 
source of true beliefs, we may say, first, that it is only 
through intellect that we know of the struggle for sur 
vival and of the biological ancestry of man : if the intel 
lect is misleading, the whole of this merely inferred history 
is presumably untrue. If, on the other hand, we agree 
with him in thinking that evolution took place as Darwin 
believed, then it is not only intellect, but all our faculties, 
that have been developed under the stress of practical 
utility. Intuition is seen at its best where it is directly 
useful, for example in regard to other people s characters 
and dispositions. Bergson apparently holds that capacity, 
for this kind of knowledge is less explicable by the 
struggle for existence than, for example, capacity for 
pure mathematics Yet the savage deceived by false 
friendship is likely to pay for his mistake with his life ; 
whereas even in the most civilised societies men are not 
put to death for mathematical incompetence. All the 
most striking of his instances of intuition in animals have 
a very direct survival value. The fact is, of course, that 
both intuition and intellect have been developed because 
they are useful, and that, speaking broadly, they are use 
ful when they give truth and become harmful when they 
give falsehood. Intellect, in civilised man, like artistic 
capacity, has occasionally been developed beyond the 
point where it is useful to the individual ; intuition, on 
the other hand, seems on the whole to diminish as 
civilisation increases. It is greater, as a rule, in children 
than in adults, in the uneducated than in the educated. 



16 MYSTICISM AND LOGIC 

Probably in dogs it exceeds anything to be found in 
human beings. But those who see in these facts a recom 
mendation of intuition ought to return to running wild 
in the woods, dyeing themselves with woad and living 
on hips and haws. 

Let us next examine whether intuition possesses any 
such infallibility as Bergson claims for it. The best 
instance of it, according to him, is our acquaintance with 
ourselves ; yet self-knowledge is proverbially rare and 
difficult. Most men, for example, have in their nature 
meannesses, vanities, and envies of which they are quite 
unconscious, though even their best friends can perceive 
them without any difficulty. It is true that intuition has 
a convincingness which is lacking to intellect : while it is 
present, it is almost impossible to doubt its truth. But 
if it should appear, on examination, to be at least as 
fallible as intellect, its greater subjective certainty be 
comes a demerit, making it only the more irresistibly 
deceptive. Apart from self-knowledge, one of the most 
notable examples of intuition is the knowledge people 
believe themselves to possess of those with whom they 
are in love : the wall between different personalities 
seems to become transparent, and people think they see 
into another soul as into their own. Yet deception in 
such cases is constantly practised with success ; and even 
where there is no intentional deception, experience 
gradually proves, as a rule, that the supposed insight 
was illusory, and that the slower more groping methods 
of the intellect are in the long run more reliable. 

Bergson maintains that intellect can only deal with 
things in so far as they resemble what has been experi 
enced in the past, while intuition has the power of appre 
hending the uniqueness and novelty that always belong 
to each fresh moment. That there is something unique 



MYSTICISM AND LOGIC 17 

and new at every moment, is certainly true ; it is also true 
that this cannot be fully expressed by means of intel 
lectual concepts. Only direct acquaintance can give 
knowledge of what is unique and new. But direct ac 
quaintance of this kind is given fully in sensation, and 
does not require, so far as I can see, any special faculty 
of intuition for its apprehension. It is neither intellect 
nor intuition, but sensation, that supplies new data ; 
but when the data are new in any remarkable manner, 
intellect is much more capable of dealing with them than 
intuition would be. The hen with a brood of ducklings 
no doubt has intuition which seems to place her inside 
them, and not merely to know them analytically ; but 
when the ducklings take to the water, the whole apparent 
intuition is seen to be illusory, and the hen is left helpless 
on the shore. Intuition, in fact, is an aspect and develop 
ment of instinct, and, like all instinct, is admirable in 
those customary surroundings which have moulded the 
habits of the animal in question, but totally incompetent 
as soon as the surroundings are changed in a way which 
demands some non-habitual mode of action. 

The theoretical understanding of the world, which is 
the aim of philosophy, is not a matter of great practical 
importance to animals, or to savages, or even to most 
civilised men. It is hardly to be supposed, therefore, 
that the rapid, rough and ready methods of instinct or 
intuition will find in this field a favourable ground for 
their application. It is the older kinds of activity, which 
bring out our kinship with remote generations of animal 
and semi-human ancestors, that show intuition at its 
best. In such matters as self-preservation and love, 
intuition will act sometimes (though not always) with a 
swiftness and precision which are astonishing to the 
critical intellect. But philosophy is not one of the 



18 MYSTICISM AND LOGIC 

pursuits which illustrate our affinity with the past : it ^ 
a highly refined, highly civilised pursuit, demanding, for 
its success, a certain liberation from the life of instinct, 
and even, at times, a certain aloofness from all mundane 
hopes and fears. It is not in philosophy, therefore, that 
we can hope to see intuition at its best. On the contrary, 
since the true objects of philosophy, and the habit of 
thought demanded for their apprehension, are strange, 
unusual, and remote, it is here, more almost than any 
where else, that intellect proves superior to intuition, 
and that quick unanalysed convictions are least deserving 
of uncritical acceptance. 

In advocating the scientific restraint and balance, as 
against the self-assertion of a confident reliance upon 
intuition, we are only urging, in the sphere of knowledge, 
that largeness of contemplation, that impersonal dis 
interestedness, and that freedom from practical pre 
occupations which have been inculcated by all the great 
religions of the world. Thus our conclusion, however it 
may conflict with the explicit beliefs of many mystics, is, 
in essence, not contrary to the spirit which inspires those 
beliefs, but rather the outcome of this very spirit as 
applied in the realm of thought. 



H. UNITY AND PLURALITY 

One of the most convincing aspects of the mystic 
illumination is the apparent revelation of the oneness of 
all things, giving rise to pantheism in religion and to 
monism in philosophy An elaborate logic, beginning 
with Parmenides, and culminating in Hegel and his 
followers, has been gradually developed, to prove that 
the universe is one indivisible Whole, and that what 
seem to be its parts, if considered as substantial and self- 



MYSTICISM AND LOGIC 19 

existing, are mere illusion. The conception of a Reality 
quite other than the world of appearance, a reality one, 
indivisible, and unchanging, was introduced into Western 
philosophy by Parmenides, not, nominally at least, for 
mystical or religious reasons, but on the basis of a logical 
argument as to the impossibility of not-being, and most 
subsequent metaphysical systems are the outcome of 
this fundamental idea. 

The logic used in defence of mysticism seems to be 
faulty as logic, and open to technical criticisms, which 1 
have explained elsewhere. I shall not here repeat these 
criticisms, since they are lengthy and difficult, but shall 
instead attempt an analysis of the state of mind from 
which mystical logic has arisen. 

Belief in a reality quite different from what appears to 
the senses arises with irresistible force in certain moods, 
which are the source of most mysticism, and of most 
metaphysics While such a mood is dominant, the need 
of logic is not felt, and accordingly the more thorough 
going mystics do not employ logic, but appeal directly 
to the immediate deliverance of their insight. But such 
fully developed mysticism is rare in the West. When 
the intensity of emotional conviction subsides, a man 
who is in the habit of reasoning will search for logical 
grounds in favour of the belief which he finds in himself. 
But since the belief already exists, he will be very hos 
pitable to any ground that suggests itself. The paradoxes 
apparently proved by his logic are really the paradoxes 
of mysticism, and are the goal which he feels his logic 
must reach if it is to be in accordance with insight. The 
resulting logic has rendered most philosophers incapable 
of giving any account of the world of science and daily 
life, If they had been anxious to give such an account, 
they would probably have discovered the errors of theii 



20 MYSTICISM AND LOGIC 

logic ; but most of them were less anxious to understand 
the world of science and daily life than to convict it of 
unreality in the interests of a super-sensible " real " 
world. 

It is in this way that logic has been pursued by those of 
the great philosophers who were mystics. But since they 
usually took for granted the supposed insight of the 
mystic emotion, their logical doctrines were presented 
with a certain dryness, and were believed by their dis 
ciples to be quite independent of the sudden illumination 
from which they sprang. Nevertheless their origin clung 
to them, and they remained to borrow a useful word 
from Mr. Santayana " malicious " in regard to the 
world of science and common sense. It is only so that 
we can account for the complacency with which philo 
sophers have accepted the inconsistency of their doctrines 
with all the common and scientific facts which seem best 
established and most worthy of belief. 

The logic of mysticism shows, as is natural, the defects 
which are inherent in anything malicious. The impulse 
to logic, not felt while the mystic mood is dominant, 
reasserts itself as the mood fades, but with a desire to 
retain the vanishing insight, or at least to prove that it 
was insight, and that what seems to contradict it is illu 
sion, The logic which thus arises is not quite dis 
interested or candid, and is inspired by a certain hatred 
of the daily world to which it is to be applied. Such an 
attitude naturally does not tend to the best results. 
Everyone knows that to read an author simply in ordei 
to refute him is not the way to understand him ; and to 
read the book of Nature with a conviction that it is all 
illusion is just as unlikely to lead to understanding. If 
our logic is to find the common world intelligible, it must 
not be hostile, but must be inspired by a genuine accept- 



MYSTICISM AND LOGIC 21 

ance such as is not usually to be found among meta 
physicians. 

III. TIME 

The unreality of time is a cardinal doctrine of many 
metaphysical systems, often nominally based, as already 
by Parmenides, upon logical arguments, but originally 
derived, at any rate in the founders of new systems, from 
the certainty which is born in the moment of mystic 
insight. As a Persian Sufi poet says : 

" Past and future are what veil God from our sight. 
Burn up both of them with fire ! How long 
Wilt thou be partitioned by these segments as a reed ? " l 

The belief that what is ultimately real must be im 
mutable is a very common one : it gave rise to the meta 
physical notion of substance, and finds, even now, a 
wholly illegitimate satisfaction in such scientific doctrines 
as the conservation of energy and mass. 

It is difficult to disentangle the truth and the error in 
this view. The arguments for the contention that time 
is unreal and that the world of sense is illusory must, I 
think, be regarded as fallacious. Nevertheless there is 
some sense easier to feel than to state in which time 
is an unimportant and superficial characteristic of reality. 
Past and future must be acknowledged to be as real as 
the present, and a certain emancipation from slavery to 
time is essential to philosophic thought. The importance 
of time is rather practical than theoretical, rather in 
relation to our desires than in relation to truth. A truer 
image of the world, I think, is obtained by picturing 
things as entering into the stream of time from an 
eternal world outside, than from a view which regards 
time as the devouring tyrant of all that is Both in 

1 Whinfield s translation of the Masnavi (Triibner, 1887), p. 34. 




aa MYSTICISM AND LOGIC 

thought and in feeling, even though time be real, to realise 
the unimportance of time is the gate of wisdom. 

That this is the case may be seen at once by asking 
ourselves why our feelings towards the past are so 
different from our feelings towards the future. The 
reason for this difference is wholly practical : our wishes 
can affect the future but not the past, the future is tc 
some extent subject to our power, while the past is un 
alterably fixed. But every future will some day be past : 
if we see the past truly now, it must, when it was still 
future, have been just what we now see it to be, and what 
is now future must be just what we shall see it to be 
when it has become past. The felt difference of quality 
between past and future, therefore, is not an intrinsic 
difference, but only a difference in relation to us : to 
impartial contemplation, it ceases to exist. And im 
partiality of contemplation is, in the intellectual sphere, 
that very same virtue of disinterestedness which, in the 
sphere of action, appears as justice and unselfishness. 
Whoever wishes to see the world truly, to rise in thought 
above the tyranny of practical desires, must learn to 
overcome the difference of attitude towards past and 
future, and to survey the whole stream of time in one 
comprehensive vision. 

The kind of way in which, as it seems to me, time ought 
not to enter into our theoretic philosophical thought, 
may be illustrated by the philosophy which has become 
associated with the idea of evolution, and which is ex 
emplified by Nietzsche, pragmatism, and Bergson. This 
philosophy, on the basis of the development which has 
led from the lowest forms of life up to man, sees in progress 
the fundamental law of the universe, and thus admits the 
difference between earlier and later into the very citadel 
of its contemplative outlook. With its past and future 



MYSTICISM AND LOGIC 23 

history of the world, conjectural as it is, I do not wish to 
quarrel. But I think that, in the intoxication of a quick 
success, much that is required for a true understanding 
of the universe has been forgotten. Something of. 
Hellenism, something, too, of Oriental resignation, must 
be combined with its hurrying Western self-assertion 
before it can emerge from the ardour of youth into the 
mature wisdom of manhood. In spite of its appeals to 
science, the true scientific philosophy, I think, is some 
thing more arduous and more aloof, appealing to less 
mundane hopes, and requiring a severer discipline for its 
successful practice. 

Darwin s Origin of Species persuaded the world that 
the difference between different species of animals and 
plants is not the fixed immutable difference that it 
appears to be. The doctrine of natural kinds, which had 
rendered classification easy and definite, which was 
enshrined in the Aristotelian tradition, and protected by 
its supposed necessity for orthodox dogma, was suddenly 
swept away for ever out of the biological world. The 
difference between man and the lower animals, which to 
our human conceit appears enormous, was shown to be a 
gradual achievement, involving intermediate being who 
could not with certainty be placed either within or with 
out the human family. The sun and the planets had 
already been shown by Laplace to be very probably 
derived from a primitive more or less undifferentiated 
nebula. Thus the old fixed landmarks became wavering 
and indistinct, and all sharp outlines were blurred. 
Things and species lost their boundaries, and none could 
say where they began or where they ended. 

But if human conceit was staggered for a moment by 
its kinship with the ape, it soon found a way to reassert 
itself, and that way is th* "philosophy" of evolution. 



24 MYSTICISM AND LOGIC 

A process which led from the amoeba to Man appeared 
to the philosophers to be obviously a progress though 
whether the amoeba would agree with this opinion is not 
known. Hence the cycle of changes which science had 
shown to be the probable history of the past was wel 
comed as revealing a law of development towards good 
in the universe an evolution or unfolding of an idea 
slowly embodying itself in the actual. But such a view, 
though it might satisfy Spencer and those whom we may 
call Hegelian evolutionists, could not be accepted as 
adequate by the more whole-hearted votaries of change. 
An ideal to which the world continuously approaches is, 
to these minds, too dead and static to be inspiring. Not 
only the aspiration, but the ideal too, must change and 
develop with the course of evolution : there must be no 
fixed goal, but a continual fashioning of fresh needs by 
the impulse which is life and which alone gives unity to 
the process. 

Life, in this philosophy, is a continuous stream, in 
which all divisions are artificial and unreal. Separate 
things, beginnings and endings, are mere convenient 
fictions : there is only smooth unbroken transition. 
The beliefs of to-day may count as true to-day, if they 
carry us along the stream ; but to-morrow they will be 
false, and must be replaced by new beliefs to meet the 
new situation. All our thinking consists of convenient 
fictions, imaginary congealings of the stream : reality 
flows on in spite of all our fictions, and though it can be 
lived, it cannot be conceived in thought. Somehow, 
without explicit statement, the assurance is slipped in 
that the future, though we cannot foresee it, will be 
better than the past or the present : the reader is like 
the child which expects a sweet because it has been told 
to oijen its mouth and shut its eyes. Logic, mathematics, 



MYSTICISM AND LOGIC 25 

physics disappear in this philosophy, because they are 
too " static " ; what is real is no impulse and movement 
towards a goal which, like the rainbow, recedes as we 
advance, and makes every place different when it reaches 
it from what it appeared to be at a distance. 

I do not propose to enter upon a technical examination 
of this philosophy. I wish only to maintain that the 
motives and interests which inspire it are so exclusively 
practical, and the problems with which it deals are so 
special, that it can hardly be regarded as touching any 
of the questions that, to my mind, constitute genuine 
philosophy. 

The predominant interest of evolutionism is in the 
question of human destiny, or at least of the destiny of 
Life. It is more interested in morality and happiness 
than in knowledge for its own sake. It must be admitted 
that the same may be said of many other philosophies, 
and that a desire for the kind of knowledge which philo 
sophy can give is very rare. But if philosophy is to 
attain truth, it is necessary first and foremost that 
philosophers should acquire the disinterested intellectual 
curiosity which characterises the genuine man of science. 
Knowledge concerning the future which is the kind of 
knowledge that must be sought if we are to know about 
human destiny is possible within certain narrow limits. 
It is impossible to say how much the limits may be en 
larged with the progress of science. But what is evident 
is that any proposition about the future belongs by its 
subject-matter to some particular science, and is to be 
ascertained/ if at all, by the methods of that science. 
Philosophy is not a short cut to the same kind of results as 
those of the other sciences : if it is to be a genuine study, 
it must have a province of its own, and aim at results 
which the other sciences can neither Drove nor disprove. 



36 MYSTICISM AND LOGIC 

Evolutionism, in basing itself upon the notion of 
progress, which is change from the worse to the better, 
allows the notion of time, as it seems to me, to become 
its tyrant rather than its servant, and thereby loses that 
impartiality of contemplation which is the source of all 
that is best in philosophic thought and feeling. Meta 
physicians, as we saw, have frequently denied altogether 
the reality of time. I do not wish to do this ; I wish 
only to preserve the mental outlook which inspired the 
denial, the attitude which, in thought, regards the past 
as having the same reality as the present and the same 
importance as the future. " In so far," says Spinoza, 1 
" as the mind conceives a thing according to the dictate 
of reason, it will be equally affected whether the idea is 
that of a future, past, or present thing." It is this " con 
ceiving according to the dictate of reason " that I find 
lacking in the philosophy which is based on evolution. 

IV. GOOD AND EVIL 

Mysticism maintains that all evil is illusory, and some 
times maintains the same view as regards good, but more 
often holds that all Reality is good. Both views are to 
be found in Heraclitus : " Good and ill are one," he says, 
but again, " To God all things are fair and good and right, 
but men hold some things wrong and some right." A 
similar twofold position is to be found in Spinoza, but he 
uses the word " perfection " when he means to speak of 
the good that is not merely human. " By reality and 
perfection I mean the same thing," he says ; 2 but else 
where we find the definition : " By good I shall mean that 
which we certainly know to be useful to us." 8 Thus 
perfection belongs to Reality in its own nature, but good- 

* Ethics. Bk. IV, Prop. LXII. Ethics. Pt. II. Df. VI. 

Ib., Pt. IV, Df. I. 



MYSTICISM AND LOGIC 27 

ness is relative to ourselves and our needs, and disappears 
in an impartial survey. Some such distinction, I think, 
is necessary in order to understand the ethical outlook 
of mysticism : there is a lower mundane kind of good 
and evil, which divides the world of appearance into 
what seem to be conflicting parts ; but there is also a 
higher, mystical kind of good, which belongs to Reality 
and is not opposed by any correlative kind of evil. 

It is difficult to give a logically tenable account of this 
position without recognising that good and evil are sub 
jective, that what is good is merely that towards which 
we have one kind of feeling, and what is evil is merely 
that towards which we have another kind of feeling. In 
our active life, where we have to exercise choice, and to 
prefer this to that of two possible acts, it is necessary to 
have a distinction of good and evil, or at least of better 
and worse. But this distinction, like everything per 
taining to action, belongs to what mysticism regards as 
the world of illusion, if only because it is essentially 
concerned with time. In our contemplative life, where 
action is not called for, it is possible to be impartial, and 
to overcome the ethical dualism which action requires. 
So long as we remain merely impartial, we may be content 
to say that both the good and the evil of action are 
illusions. But if, as we must do if we have the mystic 
vision, we find the whole world worthy of love and 
worship, if we see 

" The earth, and every common sight. . . 
Apparell d in celestial light," 

we shall say that there is a higher good than that of 
action, and that this higher good belongs to the whole 
world as it is in reality. In this way the twofold attitude 
and the apparent vacillation of mysticism are explained 
and justified. 



20 MYSTICISM AND LOGIC 

The possibility of this universal love and joy in all 
that exists is of supreme importance for the conduct and 
happiness of life, and gives inestimable value to the 
mystic emotion, apart from any creeds which may be 
built upon it. But if we are not to be led into false 
beliefs, it is necessary to realise exactly what the mystic 
emotion reveals. It reveals a possibility of human nature 
a possibility of a nobler, happier, freer life than any 
that can be otherwise achieved. But it does not reveal 
anything about the non-human, or about the nature of 
the universe in general. Good and bad, and even the 
higher good that mysticism finds everywhere, are the 
reflections of our own emotions on other things, not part 
of the substance of things as they are in themselves. 
And therefore an impartial contemplation, freed from all 
pre-occupation with Self, will not judge things good or 
bad, although it is very easily combined with that feeling 
of universal love which leads the mystic to say that the 
whole world is good. 

The philosophy of evolution, through the notion of 
progress, is bound up with the ethical dualism of the 
worse and the better, and is thus shut out, not only from 
the kind of survey which discards good and evil alto 
gether from its view, but also from the mystical belief in 
the goodness of everything. In this way the distinction 
of good and evil, like time, becomes a tyrant in this 
philosophy, and introduces into thought the restless 
selectiveness of action. Good and evil, like time, are, it 
would seem, not general or fundamental in the world of 
thought, but late and highly specialised members of the 
intellectual hierarchy. 

Although, as we saw, mysticism can be interpreted so 
as to agree with the view that good and evil are not 
intellectually fundamental, it must be admitted that here 



MYSTICISM AND LOGIC 29 

we are no longer in verbal agreement with most of the 
great philosophers and religious teachers of the past. I 
believe, however, that the elimination of ethical con 
siderations from philosophy is both scientifically necessary 
and though this may seem a paradox an ethical 
advance. Both these contentions must be briefly 
defended. 

The hope of satisfaction to our more human desires 
the hope of demonstrating that the world has this or that 
desirable ethical characteristic is not one which, so far 
as I can see, a scientific philosophy can do anything 
whatever to satisfy. The difference between a good world 
and a bad one is a difference in the particular character 
istics of the particular things that exist in these worlds : 
it is not a sufficiently abstract difference to come within 
the province of philosophy. Love and hate, for example, 
are ethical opposites, but to philosophy they are closely 
analogous attitudes towards objects. The general form 
and structure of those attitudes towards objects which 
constitute mental phenomena is a problem for philosophy, 
but the difference between love and hate is not a difference 
of form or structure, and therefore belongs rather to the 
special science of psychology than to philosophy. Thus 
the ethical interests which have often inspired philo 
sophers must remain in the background : some kind of 
ethical interest may inspire the whole study, but none 
must obtrude in the detail or be expected in the special 
results which are sought. 

If this view seems at first sight disappointing, we may 
remind ourselves that a similar change has been found 
necessary in all the other sciences. The physicist or 
chemist is not now required to prove the ethical im 
portance of his ions or atoms ; the biologist is not 
expected to prove the utility of the plants or animals 



30 MYSTICISM AND LOGIC 

which he dissects. In pre-scientific ages this was not the 
case. Astronomy, for example, was studied because 
men believed in astrology : it was thought that the 
movements of the planets had the most direct and im 
portant bearing upon the lives of human beings. Pre 
sumably, when this belief decayed and the disinterested 
study of astronomy began, many who had found astrology 
absorbingly interesting decided that astronomy had too 
little human interest to be worthy of study. Physics, as 
it appears in Plato s Timaeus for example, is full of ethical 
notions : it is an essential part of its purpose to show 
that the earth is worthy of admiration. The modern 
physicist, on the contrary, though he has no wish to deny 
that the earth is admirable, is not concerned, as physicist, 
with its ethical attributes : he is merely concerned to 
find out facts, not to consider whether they are good or 
bad. In psychology, the scientific attitude is even more 
recent and more difficult than in the physical sciences : 
it is natural to consider that human nature is either good 
or bad, and to suppose that the difference between good 
and bad, so all-important in practice, must be important 
in theory also. It is only during the last century that an 
ethically neutral psychology has grown up ; and here 
too, ethical neutrality has been essential to scientific 
success. 

In philosophy, hitherto, ethical neutrality has been 
seldom sought and hardly ever achieved. Men have 
remembered their wishes, and have judged philosophies 
in relation to their wishes. Driven from the particular 
sciences, the belief that the notions of good and evil must 
afford a key to the understanding of the world has sought 
a refuge in philosophy. But even from this last refuge, if 
philosophy is not to remain a set of pleasing dreams, this 
belief must be driven forth. It is a commonplace that 



MYSTICISM AND LOGIC 31 

happiness is not best achieved by those who seek it 
directly ; and it would seem that the same is true of the 
good. In thought, at any rate, those who forget good 
and evil and seek only to know the facts are more likely 
to achieve good than those who view the world through 
the distorting medium of their own desires. 

We are thus brought back to our seeming paradox, 
that a philosophy which does not seek to impose upon 
the world its own conceptions of good and evil is not only 
more likely to achieve truth, but is also the outcome of a 
higher ethical standpoint than one which, like evolu 
tionism and most traditional systems, is perpetually 
appraising the universe and seeking to find in it an 
embodiment of present ideals. In religion, and in every 
deeply serious view of the world and of human destiny, 
there is an element of submission, a realisation of the 
limits of human power, which is somewhat lacking in 
the modern world, with its quick material successes and 
its insolent belief in the boundless possibilities of progress. 
" He that loveth his life shall lose it " ; and there is 
danger lest, through a too confident love of life, life itself 
should lose much of what gives it its highest worth. The 
submission which religion inculcates in action is essen 
tially the same in spirit as that which science teaches in 
thought ; and the ethical neutrality by which its victories 
have been achieved is the outcome of that submission. 

The good which it concerns us to remember is the good 
which it lies in our power to create the good in our own 
lives and in our attitude towards the world. Insistence 
on belief in an external realisation of the good is a form 
of self-assertion, which, while it cannot secure the 
external good which it desires, can seriously impair the 
L> ward good which lies within our power, and destroy that 
reverence towards fact which constitutes both what is 



32 MYSTICISM AND LOGIC 

valuable in humility and what is fruitful in the scientific 
temper. 

Human beings cannot, of course, wholly transcend 
human nature ; something subjective, if only the interest 
that determines the direction of our attention, must 
remain in all our thought. But scientific philosophy 
comes nearer to objectivity than any other human 
pursuit, and gives us, therefore, the closest constant and 
the most intimate relation with the outer world that it is 
possible to achieve. To the primitive mind, everything 
is either friendly or hostile ; but experience has shown 
that friendliness and hostility are not the conceptions by 
which the world is to be understood. Scientific philo 
sophy thus represents, though as yet only in a nascent 
condition, a higher form of thought than any pre-scientific 
belief or imagination, and, like every approach to self- 
transcendence, it brings with it a rich reward in increase 
of scope and breadth and comprehension. Evolutionism, 
in spite of its appeals to particular scientific facts, fails to 
be a truly scientific philosophy because of its slavery to 
time, its ethical preoccupations, and its predominant 
interest in our mundane concerns and destiny. A truly 
scientific philosophy will be more humble, more piece 
meal, more arduous, offering less glitter of outward 
mirage to flatter fallacious hopes, but more indifferent 
to fate, and more capable of accepting the world without 
the tyrannous imposition of our human and temporary 
demands. 



II 

THE PLACE OF SCIENCE IN A 
LIBERAL EDUCATION 



SCIENCE, to the ordinary reader of newspapers, is 
represented by a varying selection of sensational 
triumphs, such as wireless telegraphy and aeroplanes 
radio-activity and the marvels of modern alchemy. It 
is not of this aspect of science that I wish to speak. 
Science, in this aspect, consists of detached up-to-date 
fragments, interesting only until they are replaced by 
something newer and more up-to-date, displaying 
nothing of the systems of patiently constructed know 
ledge out of which, almost as a casual incident, have 
come the practically useful results which interest the 
man in the street. The increased command over the 
forces of nature which is derived from science is un 
doubtedly an amply sufficient reason for encouraging 
scientific research, but this reason has been so often 
urged and is so easily appreciated that other reasons, 
to my mind quite as important, are apt to be overlooked. 
It is with these other reasons, especially with the in 
trinsic value of a scientific habit of mind in forming our 
outlook on the world, that I shall be concerned in what 
follows. 

The instance of wireless telegraphy will serve to illus 
trate the difference between the two points of view. 
Almost all th^ serious intellectual labour required for the 

33 



Jl MYSTICISM AND LOGIC 

possibility of this invention is due to three men 
Faraday, Maxwell, and Hertz. In alternating layers of 
experiment and theory these three men built up the 
modern theory of electromagnetism, and demonstrated 
the identity of light with electromagnetic waves. The 
system which they discovered is one of profound intel 
lectual interest, bringing together and unifying an end 
less variety of apparently detached phenomena, and 
displaying a cumulative mental power which cannot but 
afford delight to every generous spirit. The mechanical 
details which remained to be adjusted in order to utilise 
their discoveries for a practical system of telegraphy 
demanded, no doubt, very considerable ingenuity, but 
had not that broad sweep and that universality which 
could give them intrinsic interest as an object of dis 
interested contemplation. 

From the point of view of training the mind, of giving 
that well-informed, impersonal outlook which constitutes 
culture in the good sense of this much-misused word, it 
seems to be generally held indisputable that a literary 
education is superior to one based on science. Even the 
warmest advocates of science are apt to rest their claims 
on the contention that culture ought to be sacrificed to 
utility. Those men of science who respect culture, when 
they associate with men learned in the classics, are apt 
to admit, not merely politely, but sincerely, a certain 
inferiority on their side, compensated doubtless by the 
services which science renders to humanity, but none the 
less real. And so long as this attitude exists among men 
of science, it tends to verify itself : the intrinsically 
valuable aspects of science tend to be sacrificed to the 
merely useful, and little attempt is made to preserve that 
leisurely, systematic survey by which the finer quality 
of mind is formed and nourished. 



SCIENCE AND CULTURE 35 

But even if there be, in present fact, any such in 
feriority as is supposed in the educational value of science, 
this is, I believe, not the fault of science itself, but the 
fault of the spirit in which science is taught. If its full 
possibilities were realised by those who teach it, I believe 
that its capacity of producing those habits of mind which 
constitute the highest mental excellence would be at 
least as great as that of literature, and more particularly 
of Greek and Latin literature. In saying this I have no 
wish whatever to disparage a classical education. I have 
not myself enjoyed its benefits, and my knowledge of 
Greek and Latin authors is derived almost wholly from 
translations. But I am firmly persuaded that the Greeks 
fully deserve all the admiration that is bestowed upon 
them, and that it is a very great and serious loss to be 
unacquainted with their writings. It is not by attacking 
them, but by drawing attention to neglected excellences 
in science, that I wish to conduct my argument. 

One defect, however, does seem inherent in a purely 
classical education namely, a too exclusive emphasis 
on the past. By the study of what is absolutely ended 
and can never be renewed, a habit of criticism towards 
the present and the future is engendered. The qualities 
in which the present excels are qualities to which the 
study of the past does not direct attention, and to 
which, therefore, the student of Greek civilisation may 
easily become blind. In what is new and growing 
there is apt to be something crude, insolent, even a 
little vulgar, which is shocking to the man of sensitive 
taste ; quivering from the rough contact, he retires to 
the trim gardens of a polished past, forgetting that they 
were reclaimed from the wilderness by men as rough 
and earth-soiled as those from whom he shrinks in his 
own day. The habit of being unable to recognise merit 



36 MYSTICISM AND LOGIC 

until it is dead is too apt to be the result of a purely 
bookish life, and a culture based wholly on the past will 
seldom be able to pierce through everyday surroundings 
to the essential splendour of contemporary things, or to 
the hope of still greater splendour in the future. 

" My eyes saw not the men of old ; 
And now their age away has rolled. 
I weep to think I shall not see 
The heroes of posterity." 

So says the Chinese poet ; but such impartiality is rare 
in the more pugnacious atmosphere of the West, where 
the champions of past and future fight a never-ending 
battle, instead of combining to seek out the merits of 
both. 

This consideration, which militates not only against 
the exclusive study of the classics, but against every 
form of culture which has become static, traditional, and 
academic, leads inevitably to the fundamental ques 
tion : What is the true end of education ? But before 
attempting to answer this question it will be well to 
define the sense in which we are to use the word " educa 
tion." For this purpose I shall distinguish the sense in 
which I mean to use it from two others, both perfectly 
legitimate, the one broader and the other narrower than 
the sense in which I mean to use the word. 

In the broader sense, education will include not only 
what we learn through instruction, but all that we learn 
through personal experience the formation of character 
through the education of life. Of this aspect of education, 
vitally important as it is, I will say nothing, since its 
consideration would introduce topics quite foreign to the 
question with which we are concerned. 

In the narrower sense, education may be confined to 
instruction, the imparting of definite information on 



SCIENCE AND CULTURE 37 

various subjects, because such information, in and for 
itself, is useful in daily life. Elementary education 
reading, writing, and arithmetic is almost wholly of 
this kind. But instruction, necessary as it is, does not 
per se constitute education in the sense in which I wish 
to consider it. 

Education, in the sense in which I mean it, may be 
denned as the formation, by means of instruction, of certain 
mental habits and a certain outlook on life and the world. 
It remains to ask ourselves, what mental habits, and 
what sort of outlook, can be hoped for as the result of 
instruction ? When we have answered this question we 
can attempt to decide what science has to contribute to 
the formation of the habits and outlook which we desire. 

Our whole life is built about a certain number not a 
very small number of primary instincts and impulses. 
Only what is in some way connected with these instincts 
and impulses appears to us desirable or important ; there 
is no faculty, whether " reason " or " virtue " or what 
ever it may be called, that can take our active life and 
our hopes and fears outside the region controlled by 
these first movers of all desire. Each of them is like a 
queen-bee, aided by a hive of workers gathering honey ; 
but when the queen is gone the workers languish and 
die, and the cells remain empty of their expected sweet 
ness. So with each primary impulse in civilised man : 
it is surrounded and protected by a busy swarm of 
attendant derivative desires, which store up in its service 
whatever honey the surrounding world affords. But if 
the queen-impulse dies, the death-dealing influence, 
though retarded a little by habit, spreads slowly through 
all the subsidiary impulses, and a whole tract of life 
becomes inexplicably colourless. What was formerly 
full of zest, and so obviously worth doing that it raised 



3* MYSTICISM AND LOGIC 

no questions, has now grown dreary and purposeless : 
with a sense of disillusion we inquire the meaning of life, 
and decide, perhaps, that all is vanity. The search for 
an outside meaning that can compel an inner response 
must always be disappointed : all " meaning " must be 
at bottom related to our primary desires, and when they 
are extinct no miracle can restore to the world the value 
which they reflected upon it. 

The purpose of education, therefore, cannot be to 
create any primary impulse which is lacking in the 
uneducated ; the purpose can only be to enlarge the 
scope of those that human nature provides, by increasing 
the number and variety of attendant thoughts, and by 
showing where the most permanent satisfaction is to be 
found. Under the impulse of a Calvinistic horror of 
the " natural man," this obvious truth has been too 
often misconceived in the training of the young ; 
" nature " has been falsely regarded as excluding all 
that is best in what is natural, and the endeavour to 
teach virtue has led to the production of stunted and 
contorted hypocrites instead of full-grown human beings. 
From such mistakes in education a better psychology or 
a kinder heart is beginning to preserve the present 
generation ; we need, therefore, waste no more words on 
the theory that the purpose of education is to thwart or 
eradicate nature. 

But although nature must supply the initial force of 
desire, nature is not, in the civilised man, the spasmodic, 
fragmentary, and yet violent set of impulses that it is 
in the savage. Each impulse has its constitutional 
ministry of thought and knowledge and reflection, 
through which possible conflicts of impulses are foreseen, 
and temporary impulses are controlled by the unifying 
impulse which may be called wisdom. In this way 



SCIENCE AND CULTURE 39 

education destroys the crudity of instinct, and increases 
through knowledge the wealth and variety of the indi 
vidual s contacts with the outside world, making him 
no longer an isolated righting unit, but a citizen of the 
universe, embracing distant countries, remote regions of 
space, and vast stretches of past and future within the 
circle of his interests. It is this simultaneous softening 
in the insistence of desire and enlargement of its scope 
that is the chief moral end of education. 

Closely connected with this moral end is the more 
purely intellectual aim of education, the endeavour to 
make us see and imagine the world in an objective 
manner, as far as possible as it is in itself, and not merely 
through the distorting medium of personal desire. The 
complete attainment of such an objective view is no 
doubt an ideal, indefinitely approachable, but not actually 
and fully realisable. Education, considered as a process 
of forming our mental habits and our outlook on the 
world, is to be judged successful in proportion as its out 
come approximates to this ideal ; in proportion, that is 
to say, as it gives us a true view of our place in society, 
of the relation of the whole human society to its non- 
human environment, and of the nature of the non- 
human world as it is in itself apart from our desires and 
interests. If this standard is admitted, we can return 
to the consideration of science, inquiring how far science 
contributes to such an aim, and whether it is in any 
respect superior to its rivals in educational practice. 



II 

Two opposite and at first sight conflicting merit* 
belong to science as against literature and art. The one, 
which is not inherently necessary, but is certainly true 



40 MYSTICISM AND LOGIC 

at the present day, is hopefulness as to the future of 
human achievement, and in particular as to the useful 
work that may be accomplished by any intelligent 
student. This merit and the cheerful outlook which it 
engenders prevent what might otherwise be the de 
pressing effect of another aspect of science, to my mind 
also a merit, and perhaps its greatest merit I mean the 
irrelevance of human passions and of the whole subjective 
apparatus where scientific truth is concerned. Each of 
these reasons for preferring the study of science requires 
some amplification. Let us begin with the first. 

In the study of literature or art our attention is per 
petually riveted upon the past : the men of Greece or 
of the Renaissance did better than any men do now ; the 
triumphs of former ages, so far from facilitating fresh 
triumphs in our own age, actually increase the diffi 
culty of fresh triumphs by rendering originality harder 
of attainment ; not only is artistic achievement not 
cumulative, but it seems even to depend upon a certain 
freshness and naivete of impulse and vision which civilisa 
tion tends to destroy. Hence comes, to those who have 
been nourished on the literary and artistic productions 
of former ages, a certain peevishness and undue fas 
tidiousness towards the present, from which there 
seems no escape except into the deliberate vandalism 
which ignores tradition and in the search after originality 
achieves only the eccentric. But in such vandalism 
there is none of the simplicity and spontaneity out of 
which great art springs : theory is still the canker in its 
core, and insincerity destroys the advantages of a merely 
pretended ignorance. 

The despair thus arising from an education which 
suggests no pre-eminent mental activity except that of 
artistic creation is wholly absent from an education 



SCIENCE AND CULTURE 41 

which gives the knowledge of scientific method. The 
discovery of scientific method, except in pure mathe 
matics, is a thing of yesterday ; speaking broadly, we 
may say that it dates from Galileo. Yet already it has 
transformed the world, and its success proceeds with 
ever-accelerating velocity. In science men have dis 
covered an activity of the very highest value in which 
they are no longer, as in art, dependent for progress 
upon the appearance of continually greater genius, for 
in science the successors stand upon the shoulders of 
their predecessors ; where one man of supreme genius 
has invented a method, a thousand lesser men can apply 
it. No transcendent ability is required in order to make 
useful discoveries in science ; the edifice of science needs 
its masons, bricklayers, and common labourers as well 
as its foremen, master-builders, and architects. In art 
nothing worth doing can be done without genius ; in 
science even a very moderate capacity can contribute to 
a supreme achievement. 

In science the man of real genius is the man who 
invents a new method. The notable discoveries are 
often made by his successors, who can apply the method 
with fresh vigour, unimpaired by the previous labour of 
perfecting it ; but the mental calibre of the thought 
required for their work, however brilliant, is not so great 
as that required by the first inventor of the method. 
There are in science immense numbers of different 
methods, appropriate to different classes of problems ; 
but over and above them all, there is something not 
easily definable, which may be called the method of 
science. It was formerly customary to identify this 
with the inductive method, and to associate it with the 
name of Bacon. But the true inductive method was 
not discovered by Bacon, and the true method of science 



42 MYSTICISM AND LOGIC 

is something which includes deduction as mucn as 
induction, logic and mathematics as much as botany and 
geology. I shall not attempt the difficult task of stating 
what the scientific method is, but I will try to indicate 
the temper of mind out of which the scientific method 
grows, which is the second of the two merits that were 
mentioned above as belonging to a scientific education. 

The kernel of the scientific outlook is a thing so simple, 
so obvious, so seemingly trivial, that the mention of it 
may almost excite derision. The kernel of the scientific 
outlook is the refusal to regard our own desires, tastes, 
and interests as affording a key to the understanding of 
the world. Stated thus baldly, this may seem no more 
than a trite truism. But to remember it consistently in 
matters arousing our passionate partisanship is by no 
means easy, especially where the available evidence is 
uncertain and inconclusive. A few illustrations will 
make this clear. 

Aristotle, I understand, considered that the stars 
must move in circles because the circle is the most 
perfect curve. In the absence of evidence to the con 
trary, he allowed himself to decide a question of fact by 
an appeal to aesthetico-moral considerations. In such 
a case it is at once obvious to us that this appeal was 
unjustifiable. We know now how to ascertain as a fact 
the way in which the heavenly bodies move, and we 
know that they do not move in circles, or even in 
accurate ellipses, or in any other kind of simply de- 
scribable curve. This may be painful to a certain 
hankering after simplicity of pattern in the universe, 
but we know that in astronomy such feelings are irre 
levant. Easy as this knowledge seems now, we owe it 
to the courage and insight of the first inventors of scien 
tific method, and more especially of Galileo. 



SCIENCE AND CULTURE 43 

We may take as another illustration Malthus s 
doctrine of population. This illustration is all the better 
for the fact that his actual doctrine is nov? known to be 
largely erroneous. It is not his conclusions that are 
valuable, but the temper and method of his inquiry 
As everyone knows, it was to him that Darwin owed ar 
essential part of his theory of natural selection, ana 
this was only possible because Malthus s outlook was 
truly scientific. His great merit lies in considering man 
not as the object of praise or blame, but as a part of 
nature, a thing with a certain characteristic behaviour 
from which certain consequences must follow. If the 
behaviour is not quite what Malthus supposed, if the 
consequences are not quite what he inferred, that may 
falsify his conclusions, but does not impair the value of 
his method. The objections which were made when his 
doctrine was new that it was horrible and depressing, 
that people ought not to act as he said they did, and so 
on were all such as implied an unscientific attitude of 
mind ; as against all of them, his calm determination 
to treat man as a natural phenomenon marks an im 
portant advance over the reformers of the eighteenth 
century and the Revolution. 

Under the influence of Darwinism the scientific atti 
tude towards man has now become fairly common, and 
is to some people quite natural, though to most it is still a 
difficult and artificial intellectual contortion. There is 
however, one study which is as yet almost wholly un 
touched by the scientific spirit I mean the study of 
philosophy Philosophers and the public imagine that 
the scientific spirit must pervade pages that bristle with 
allusions to ions, germ-plasms, and the eyes of shell-fish. 
But as the devil can quote Scripture, so the philosopher 
can quote science. The scientific spirit is not an affair of. 



44 MYSTICISM AND LOGIC 

quotation, of externally acquired information, any more 
than manners are an affair of the etiquette-book. The 
scientific attitude of mind involves a sweeping away of 
all other desires in the interests of the desire to know 
it involves suppression of hopes and fears, loves and 
hates, and the whole subjective emotional life, until we 
become subdued to the material, able to see it frankly, 
without preconceptions, without bias, without any wish 
except to see it as it is, and without any belief that what 
it is must be determined by some relation, positive or 
negative, to what we should like it to be, or to what we 
can easily imagine it to be. 

Now in philosophy this attitude of mind has not as 
yet been achieved. A certain self-absorption, not per 
sonal, but human, has marked almost all attempts to 
conceive the universe as a whole. Mind, or some aspect 
of it thought or will or sentience has been regarded 
as the pattern after which the universe is to be con 
ceived, for no better reason, at bottom, than that such 
a universe would not seem strange, and would give 
us the cosy feeling that every place is like home. To 
conceive the universe as essentially progressive or essen 
tially deteriorating, for example, is to give to our hopes 
and fears a cosmic importance which may, of course, 
be justified, but which we have as yet no reason to suppose 
justified. Until we have learnt to think of it in ethically 
neutral terms, we have not arrived at a scientific attitude 
in philosophy ; and until we have arrived at such an 
attitude, it is hardly to be hoped that philosophy will 
achieve any solid results. 

I have spoken so far largely of the negative aspect of the 
scientific spirit, but it is from the positive aspect that its 
value is derived. The instinct of constructiveness, which is 
one of the chief incentives to artistic creation, can find 



SCIENCE AND CULTURE 45 

in scientific systems a satisfaction more massive than 
any epic poem. Disinterested curiosity, which is the 
source of almost all intellectual effort, finds with aston 
ished delight that science can unveil secrets which 
might well have seemed for ever undiscoverable. The 
desire for a larger life and wider interests, for an escape 
from private circumstances, and even from the whole 
recurring human cycle of birth and death, is fulfilled by 
the impersonal cosmic outlook of science as by nothing 
else. To all these must be added, as contributing to the 
happiness of the man of science, the admiration of 
splendid achievement, and the consciousness of inestim 
able utility to the human race. A life devoted to science 
is therefore a happy life, and its happiness is derived 
from the very best sources that are open to dwellers on 
this troubled and passionate planet. 



Ill 

A FREE MAN S WORSHIP 1 

TO Dr. Faustus in his study Mephistopheles told the 
history of the Creation, saying : 

" The endless praises of the choirs of angels had begun 
to grow wearisome ; for, after all, did he not deserve 
their praise ? Had he not given them endless joy ? 
Would it not be more amusing to obtain undeserved 
praise, to be worshipped by beings whom he tortured ? 
He smiled inwardly, and resolved that the great drama 
should be performed. 

" For countless ages the hot nebula whirled aimlessly 
through space. At length it began to take shape, the 
central mass threw off planets, the planets cooled, boil 
ing seas and burning mountains heaved and tossed, 
from black masses of cloud hot sheets of rain deluged 
the barely solid crust. And now the first germ of life 
grew in the depths of the ocean, and developed rapidly 
in the fructifying warmth into vast forest trees, huge 
ferns springing from the damp mould, sea monsters 
breeding, fighting, devouring, and passing away. And 
from the monsters, as the play unfolded itself, Man was 
born, with the power of thought, the knowledge of good 
and evil, and the cruel thirst for worship. And Man 
saw that all is passing in this mad, monstrous world, 
that all is struggling to snatch, at any cost, a few brief 
moments of life before Death s inexorable decree. And 

1 Reprinted from the Independent Review, December, 1903. 
46 



A FREE MAN S WORSHIP 47 

Man said : There is a hidden purpose, could we but 
fathom it, and the purpose is good ; for we must rever 
ence something, and in the visible world there is nothing 
worthy of reverence/ And Man stood aside from the 
struggle, resolving that God intended harmony to come 
out of chaos by human efforts. And when he followed 
the instincts which God had transmitted to him from 
his ancestry of beasts of prey, he called it Sin, and asked 
God to forgive him. But he doubted whether he could 
be justly forgiven, until he invented a divine Plan by 
which God s wrath was to have been appeased. And 
seeing the present was bad, he made it yet worse, that 
thereby the future might be better. And he gave God 
thanks for the strength that enabled him to forgo even 
the joys that were possible. And God smiled ; and 
when he saw that Man had become perfect in renuncia 
tion and worship, he sent another sun through the sky, 
which crashed into Man s sun ; and all returned again 
to nebula. 

Yes, he murmured, it was a good play ; I will 
have it performed again. 

Such, in outline, but even more purposeless, more 
void of meaning, is the world which Science presents for 
our belief. Amid such a world, if anywhere, our ideals 
henceforward must find a home. That Man is the 
product of causes which had no prevision of the end 
they were achieving ; that his origin, his growth, his 
hopes and fears, his loves and his beliefs, are but the 
outcome of accidental collocations of atoms ; that no fire, 
no heroism, no intensity of thought and feeling, can 
preserve an individual life beyond the grave ; that all 
the labours of the ages, all the devotion, all the inspira 
tion, all the noonday brightness of human genius, are 
destined to extinction in the vast death of the solar 



48 MYSTICISM AND LOGIC 

system, and that the whole temple of Man s achieve 
ment must inevitably be buried beneath the d6bris of a 
universe in ruins all these things, if not quite beyond 
dispute, are yet so nearly certain, that no philosophy 
which rejects them can hope to stand. Only within 
the scaffolding of these truths, only on the firm founda 
tion of unyielding despair, can the soul s habitation 
henceforth be safely built. 

How, in such an alien and inhuman world, can so 
powerless a creature as Man preserve his aspirations 
untarnished ? A strange mystery it is that Nature, 
omnipotent but blind, in the revolutions of her secular 
hurryings through the abysses of space, has brought 
forth at last a child, subject still to her power, but 
gifted with sight, with knowledge of good and evil, with 
the capacity of judging all the works of his unthinking 
Mother. In spite of Death, the mark and seal of the 
parental control, Man is yet free, during his brief years, 
to examine, to criticise, to know, and in imagination to 
create. To him alone, in the world with which he is 
acquainted, this freedom belongs ; and in this lies his 
superiority to the resistless forces that control his out 
ward life. 

The savage, like ourselves, feels the oppression of his 
impotence before the powers of Nature ; but having in 
himself nothing that he respects more than Power, he is 
willing to prostrate himself before his gods, without 
inquiring whether they are worthy of his worship. 
Pathetic and very terrible is the long history of cruelty 
and torture, of degradation and human sacrifice, endured 
in the hope of placating the jealous gods : surely, the 
trembling believer thinks, when what is most precious 
has been freely given, their lust for blood must be ap 
peased, and more will not be required. The religion of 



A FREE MAN S WORSHIP 49 

Moloch as such creeds may be generically called is in 
essence the cringing submission of the slave, who dare 
not, even in his heart, allow the thought that his master 
deserves no adulation. Since the independence of ideals 
is not yet acknowledged, Power may be freely wor 
shipped, and receive an unlimited respect, despite its 
wanton infliction of pain. 

But gradually, as morality grows bolder, the claim of 
the ideal world begins to be felt ; and worship, if it is 
not to cease, must be given to gods of another kind than 
those created by the savage. Some, though they feel 
the demands of the ideal, will stih 1 consciously reject 
them, still urging that naked Power is worthy of worship. 
Such is the attitude inculcated in God s answer to Job 
out of the whirlwind : the divine power and knowledge 
are paraded, but of the divine goodness there is no hint. 
Such also is the attitude of those who, in our own day, 
base their morality upon the struggle for survival, main 
taining that the survivors are necessarily the fittest. 
But others, not content with an answer so repugnant to 
the moral sense, will adopt the position which we have 
become accustomed to regard as specially religious, 
maintaining that, in some hidden manner, the world of 
fact is really harmonious with the world of ideals. Thus 
Man creates God, all-powerful and all-good, the mystic 
unity of what is and what should be. 

But the world of fact, after all, is not good ; and, in 
submitting our judgment to it, there is an element of 
slavishness from which our thoughts must be purged. 
For in all things it is well to exalt the dignity of Man, 
by freeing him as far as possible from the tyranny of 
non-human Power. When we have realised that Power 
is largely bad, that man, with his knowledge of good and 
evil, is but a helpless atom in a world which has no such 



50 MYSTICISM AND LOGIC 

knowledge, the choice is again presented to us : Shall 
we worship Force, or shall we worship Goodness ? Shall 
our God exist and be evil, or shall he be recognised as 
the creation of our own conscience ? 

The answer to this question is very momentous, and 
affects profoundly our whole morality. The worship of 
Force, to which Carlyle and Nietzsche and the creed of 
Militarism have accustomed us, is the result of failure to 
maintain our own ideals against a hostile universe : it is 
itself a prostrate submission to evil, a sacrifice of our 
best to Moloch. If strength indeed is to be respected, 
let us respect rather the strength of those who refuse 
that false " recognition of facts " which fails to recog 
nise that facts are often bad. Let us admit that, in the 
world we know, there are many things that would be 
better otherwise, and that the ideals to which we do and 
must adhere are not realised in the realm of matter. Let 
us preserve our respect for truth, for beauty, for the 
ideal of perfection which life does not permit us to 
attain, though none of these things meet with the ap 
proval of the unconscious universe. If Power is bad, as 
it seems to be, let us reject it from our hearts. In this 
lies Man s true freedom : in determination to worship 
only the God created by our own love of the good, to 
respect only the heaven which inspires the insight of our 
best moments. In action, in desire, we must submit 
perpetually to the tyranny of outside forces j/ but in 
thought, in aspiration, we are free, free from our fellow- 
men, free from the petty planet on which our bodies 
impotently crawl, free even, while we live, from the 
tyranny of death. Let us learn, then, that energy of 
faith which enables us to live constantly in the vision of 
the good ; and let us descend, in action, into the world 
of *act, with that vision always before us. 



A FREE MAN S WORSHIP 51 

When first the opposition of fact and ideal grows fully 
visible, a spirit of fiery revolt, of fierce hatred of the gods, 
seems necessary to the assertion of freedom. To defy 
with Promethean constancy a hostile universe, to keep 
its evil always in view, always actively hated, to refuse 
no pain that the malice of Power can invent, appears to 
be the duty of all who will not bow before the inevitable. 
But indignation is still a bondage, for it compels our 
thoughts to be occupied with an evil world ; and in the 
fierceness of desire from which rebellion springs there is 
a kind of self-assertion which it is necessary for the wise 
to overcome. Indignation is a submission of our thoughts, 
but not of our desires ; the Stoic freedom in which 
wisdom consists is found in the submission of our desires, 
but not of our thoughts. From the submission of our 
desires springs the virtue of resignation ; from the free 
dom of our thoughts springs the whole world of art and 
philosophy, and the vision of beauty by which, at last, 
we half reconquer the reluctant world. But the vision 
of beauty is possible only to unfettered contemplation, 
to thoughts not weighted by the load of eager wishes ; 
and thus Freedom comes only to those who no longer 
ask of life that it shall yield them any of those personal 
goods that are subject to the mutations of Time. 

Although the necessity of renunciation is evidence of 
the existence of evil, yet Christianity, in preaching it, 
has shown a wisdom exceeding that of the Promethean 
philosophy of rebellion. It must be admitted that, of 
the things we desire, some, though they prove impossible, 
are yet real goods ; others, however, as ardently longed 
for, do not form part of a fully purified ideal. The belief 
that what must be renounced is bad, though sometimes 
false, is far less often false than untamed passion sup 
poses ; and the creed of religion, by providing a reason 



52 MYSTICISM AND LOGIC 

for proving that it is never false, has been the means of 
purifying our hopes by the discovery of many austere 
truths. 

But there is in resignation a further good element : 
even real goods, when they are unattainable, ought not 
to be fretfully desired. To every man comes, sooner or 
later, the great renunciation. For the young, there is 
nothing unattainable ; a good thing desired with the 
whole force of a passionate will, and yet impossible, is to 
them not credible. Yet, by death, by illness, by poverty, 
or by the voice of duty, we must learn, each one of us, 
that the world was not made for us, and that, however 
beautiful may be the things we crave, Fate may never 
theless forbid them. It is the part of courage, when mis 
fortune comes, to bear without repining the ruin of our 
hopes, to turn away our thoughts from vain regrets. 
This degree of submission to Power is not only just and 
right : it is the very gate of wisdom. 

But passive renunciation is not the whole of wisdom ; 
for not by renunciation alone can we build a temple for 
the worship of our own ideals. Haunting foreshado wings 
of the temple appear in the realm of imagination, in 
music, in architecture, in the untroubled kingdom of 
reason, and in the golden sunset magic of lyrics, where 
beauty shines and glows, remote from the touch of 
sorrow, remote from the fear of change, remote from the 
failures and disenchantments of the world of fact. In 
the contemplation of these things the vision of heaven 
will shape itself in our hearts, giving at once a touch 
stone to judge the world about us, and an inspiration by 
which to fashion to our needs whatever is not incapable 
of serving as a stone in the sacred temple. 

Except for those rare spirits that are born without sin, 
there is a cavern of darkness to be traversed before that 



A FREE MAN S WORSHIP 53 

temple can be entered. The gate of the cavern is despair, 
and its floor is paved with the gravestones of abandoned 
hopes. There Self must die ; there the eagerness, the 
greed of untamed desire must be slain, for only so can 
the soul be freed from the empire of Fate. But out of 
the cavern the Gate of Renunciation leads again to the 
daylight of wisdom, by whose radiance a new insight, a 
new joy, a new tenderness, shine forth to gladden the 
pilgrim s heart. 

When, without the bitterness of impotent rebellion, 
we have learnt both to resign ourselves to the outward 
rule of Fate and to recognise that the non-human world 
is unworthy of our worship, it becomes possible at last 
so to transform and refashion the unconscious universe, 
so to transmute it in the crucible of imagination, that a 
new image of shining gold replaces the old idol of clay. 
In all the multiform facts of the world in the visual 
shapes of trees and mountains and clouds, in the events 
of the life of man, even in the very omnipotence of Death 
the insight of creative idealism can find the reflection 
of a beauty which its own thoughts first made. In this 
way mind asserts its subtle mastery over the thoughtless 
forces of Nature. The more evil the material with which 
it deals, the more thwarting to untrained desire, the 
greater is its achievement in inducing the reluctant rock 
to yield up its hidden treasures, the prouder its victory 
in compelling the opposing forces to swell the pageant of 
its triumph. Of all the arts, Tragedy is the proudest, the 
most triumphant ; for it builds its shining citadel in the 
very centre of the enemy s country, on the very summit 
of his highest mountain ; from its impregnable watch- 
towers, his camps and arsenals, his columns and forts, 
are all revealed ; within its walls the free life continues, 
while the legions of Death and Pain and Despair, and all 



54 MYSTICISM AND LOGIC 

the servile captains of tyrant Fate, afford the burghers 
of that dauntless city new spectacles of beauty. Happy 
those sacred ramparts, thrice happy the dwellers on that 
all-seeing eminence. Honour to those brave warriors 
who, through countless ages of warfare, have preserved 
for us the priceless heritage of liberty, and have kept 
undefiled by sacrilegious invaders the home of the un 
subdued. 

But the beauty of Tragedy does but make visible a 
quality which, in more or less obvious shapes, is present 
always and everywhere in life. In the spectacle of Death, 
in the endurance of intolerable pain, and in the irrevocable- 
ness of a vanished past, there is a sacredness, an over 
powering awe, a feeling of the vastness, the depth, the 
inexhaustible mystery of existence, in which, as by some 
strange marriage of pain, the sufferer is bound to the 
world by bonds of sorrow. In these moments of insight, 
we lose all eagerness of temporary desire, all struggling 
and striving for petty ends, all care for the little trivial 
things that, to a superficial view, make up the common 
life of day by day ; we see, surrounding the narrow raft 
illumined by the flickering light of human comradeship, 
the dark ocean on whose rolling waves we toss for a brief 
hour ; from the great night without, a chill blast breaks 
in upon our refuge ; all the loneliness of humanity amid 
hostile forces is concentrated upon the individual soul, 
which must struggle alone, with what of courage it can 
command, against the whole weight of a universe that 
cares nothing for its hopes and fears. Victory, in this 
struggle with the powers of darkness, is the true baptism 
into the glorious company of heroes, the true initiation 
into the overmastering beauty of human existence. From 
that awful encounter of the soul with the outer world, 
enunciation, wisdom, and charity are born ; and with 



A FREE MAN S WORSHIP 55 

their birth a new life begins. To take into the inmost 
shrine of the soul the irresistible forces whose puppets 
we seem to be Death and change, the irrevocableness 
of the past, and the powerlessness of man before 
the blind hurry of the universe from vanity to vanity 
to feel these things and know them is to conquer 
them. 

This is the reason why the Past has such magical 
power. The beauty of its motionless and silent pictures 
is like the enchanted purity of late autumn, when the 
leaves, though one breath would make them fall, still 
glow against the sky in golden glory. The Past does not 
change or strive ; like Duncan, after life s fitful fever it 
sleeps well ; what was eager and grasping, what was 
petty and transitory, has faded away, the things that 
were beautiful and eternal shine out of it like stars in the 
night. Its beauty, to a soul not worthy of it, is un 
endurable ; but to a soul which has conquered Fate it is 
the key of religion. 

The life of Man, viewed outwardly, is but a small 
thing in comparison with the forces of Nature. The 
slave is doomed to worship Time and Fate and Death, 
because they are greater than anything he finds in him 
self, and because all his thoughts are of things which 
they devour. But, great as they are, to think of them 
greatly, to feel their passionless splendour, is greater 
still. And such thought makes us free men ; we no 
longer bow before the inevitable in Oriental subjection, 
but we absorb it, and make it a part of ourselves. To 
abandon the struggle for private happiness, to expel all 
eagerness of temporary desire, to burn with passion for 
eternal things this is emancipation, and this is the free 
man s worship. And this liberation is effected by a con 
templation of Fate ; for Fate itself is subdued by the 



5 6 MYSTICISM AND LOGIC 

mind which leaves nothing to be purged by the purifying 
fire of Time. 

United with his fellow-men by the strongest of all ties, 
the tie of a common doom, the free man finds that a new 
vision is with him always, shedding over every daily 
task the light of love. The life of Man is a long march 
through the night, surrounded by invisible foes, tortured 
by weariness and pain, towards a goal that few can hope 
to reach, and where none may tarry long. One by one, 
as they march, our comrades vanish from our sight, 
seized by the silent orders of omnipotent Death. Very 
brief is the time in which we can help them, in which 
their happiness or misery is decided. Be it ours to shed 
sunshine on their path, to lighten their sorrows by the 
balm of sympathy, to give them the pure joy of a never- 
tiring affection, to strengthen failing courage, to instil 
faith in hours of despair. Let us not weigh in grudging 
scales their merits and demerits, but let us think only of 
their need of the sorrows, the difficulties, perhaps the 
blindnesses, that make the misery of their lives ; let us 
remember that they are fellow-sufferers in the same 
darkness, actors in the same tragedy with ourselves. 
And so, when their day is over, when their good and 
their evil have become eternal by the immortality of the 
past, be it ours to feel that, where they suffered, where 
they failed, no deed of ours was the cause ; but wherever 
a spark of the divine fire kindled in their hearts, we were 
ready with encouragement, with sympathy, with brave 
words in which high courage glowed. 

Brief and powerless is Man s life ; on him and all his 
race the slow, sure doom falls pitiless and dark. Blind 
to good and evil, reckless of destruction, omnipotent 
matter rolls on its relentless way ; for Man, condemned 
to-day to lose his dearest, to-morrow himself to pass 



A FREE MAN S WORSHIP 57 

through the gate of darkness, it remains only to cherish, 
ere yet the blow falls, the lofty thoughts that ennoble 
his little day ; disdaining the coward terrors of the slave 
of Fate, to worship at the shrine that his own hands have 
built ; undismayed by the empire of chance, to preserve 
a mind free from the wanton tyranny that rules his out 
ward life ; proudly defiant of the irresistible forces that 
tolerate, for a moment, his knowledge and his condemna 
tion, to sustain alone, a weary but unyielding Atlas, the 
world that his own ideals have fashioned despite the 
trampling march of unconscious power, 



IV 

THE STUDY OF MATHEMATICS 

IN regard to every form of human activity it is neces 
sary that the question should be asked from time to 
time, What is its purpose and ideal ? In what way does 
it contribute to the beauty of human existence ? As 
respects those pursuits which contribute only remotely, 
by providing the mechanism of life, it is well to be 
reminded that not the mere fact of living is to be desired, 
but the art of living in the contemplation of great things. 
Still more in regard to those avocations which have no 
end outside themselves, which are to be justified, if at all, 
as actually adding to the sum of the world s permanent 
possessions, it is necessary to keep alive a knowledge of 
their aims, a clear prefiguring vision of the temple in 
which creative imagination is to be embodied. 

The fulfilment of this need, in what concerns the 
studies forming the material upon which custom has 
decided to train the youthful mind, is indeed sadly 
remote so remote as to make the mere statement of 
such a claim appear preposterous. Great men, fully 
alive to the beauty of the contemplations to whose 
service their lives are devoted, desiring that others may 
share in their joys, persuade mankind to impart to the 
successive generations the mechanical knowledge with 
out which it is impossible to cross the threshold. Dry 
pedants possess themselves of the privilege of instilling 
this knowledge : they forget that it is to serve but as a 



THE STUDY OF MATHEMATICS 59 

key to open the doors of the temple ; though they spend 
their lives on the steps leading up to those sacred doors, 
they turn their backs upon the temple so resolutely that 
its very existence is forgotten, and the eager youth, who 
would press forward to be initiated to its domes and 
arches, is bidden to turn back and count the steps. 

Mathematics, perhaps more even than the study of 
Greece and Rome, has suffered from this oblivion of its 
due place in civilisation. Although tradition has decreed 
that the great bulk of educated men shall know at least 
the elements of the subject, the reasons for which the 
tradition arose are forgotten, buried beneath a great 
rubbish-heap of pedantries and trivialities. To those 
who inquire as to the purpose of mathematics, the usual 
answer will be that it facilitates the making of machines, 
the travelling from place to place, and the victory over 
foreign nations, whether in war or commerce. If it be 
objected that these ends all of which are of doubtful 
value are not furthered by the merely elementary 
study imposed upon those who do not become expert 
mathematicians, the reply, it is true, will probably be 
that mathematics trains the reasoning faculties. Yet 
the very men who make this reply are, for the most part, 
unwilling to abandon the teaching of definite fallacies, 
known to be such, and instinctively rejected by the un 
sophisticated mind of every intelligent learner. And the 
reasoning faculty itself is generally conceived, by those 
who urge its cultivation, as merely a means for the avoid 
ance of pitfalls and a help in the discovery of rules for 
the guidance of practical life. All these are undeniably 
important achievements to the credit of mathematics ; 
yet it is none of these that entitles mathematics to a place 
in every liberal education. Plato, we know, regarded the 
contemplation of mathematical truths as worthy of the 



60 MYSTICISM AND LOGIC 

Deity ; and Plato realised, more perhaps than any other 
single man, what those elements are in human life which 
merit a place in heaven. There is in mathematics, he 
says, "something which is necessary and cannot be set 
aside . . . and, if I mistake not, of divine necessity ; for 
as to the human necessities of which the Many talk in 
this connection, nothing can be more ridiculous than such 
an application of the words. Cleinias. And what are these 
necessities of knowledge, Stranger, which are divine and 
not human ? Athenian. Those things without some use 
or knowledge of which a man cannot become a God to 
the world, nor a spirit, nor yet a hero, nor able earnestly 
to think and care for man " (Laws, p. SiS). 1 Such was 
Plato s judgment of mathematics ; but the mathe 
maticians do not read Plato, while those who read him 
know no mathematics, and regard his opinion upon this 
question as merely a curious aberration. 

Mathematics, rightly viewed, possesses not only truth, 
but supreme beauty a beauty cold and austere, like 
that of sculpture, without appeal to any part of our 
weaker nature, without the gorgeous trappings of paint 
ing or music, yet sublimely pure, and capable of a stern 
perfection such as only the greatest art can show. The 
true spirit of delight, the exaltation, the sense of being 
more than man, which is the touchstone of the highest 
excellence, is to be found in mathematics as surely as in 
poetry. What is best in mathematics deserves not merely 
to be learnt as a task, but to be assimilated as a part of 
daily thought, and brought again and again before the 
mind with ever-renewed encouragement. Real life is, to 
most men, a long second-best, a perpetual compromise 
between the ideal and the possible ; but the world of 
pure reason knows no compromise, no practical limita- 

1 This passage was pointed out to me by Professor Gilbert Murray. 



THE STUDY OF MATHEMATICS 61 

tions, no barrier to the creative activity embodying in 
splendid edifices the passionate aspiration after the per 
fect from which all great work springs. Remote from 
human passions, remote even from the pitiful facts of 
nature, the generations have gradually created an 
ordered cosmos, where pure thought can dwell as in its 
natural home, and where one, at least, of our nobler 
impulses can escape from the dreary exile of the actual 
world. 

So little, however, have mathematicians aimed at 
beauty, that hardly anything in their work has had this 
conscious purpose. Much, owing to irrepressible instincts, 
which were better than avowed beliefs, has been moulded 
by an unconscious taste ; but much also has been spoilt 
by false notions of what was fitting. The characteristic 
excellence of mathematics is only to be found where the 
reasoning is rigidly logical : the rules of logic are to 
mathematics what those of structure are to architecture. 
In the most beautiful work, a chain of argument is pre 
sented in which every link is important on its own 
account, in which there is an air of ease and lucidity 
throughout, and the premises achieve more than would 
have been thought possible, by means which appeal- 
natural and inevitable. Literature embodies what is 
general in particular circumstances whose universal 
significance shines through their individual dress ; but 
mathematics endeavours to present whatever is most 
general in its purity, without any irrelevant trappings. 

How should the teaching of mathematics be conducted 
so as to communicate to the learner as much as possible 
of this high ideal ? Here experience must, in a great 
measure, be our guide ; but some maxims may result 
from our consideration of the ultimate purpose to be 
achieved. 



6a MYSTICISM AND LOGIC 

One of the chief ends served by mathematics, when 
rightly taught, is to awaken the learner s belief in reason, 
his confidence in the truth of what has been demon 
strated, and in the value of demonstration. This purpose 
is not served by existing instruction ; but it is easy to 
see ways in which it might be served. At present, in 
what concerns arithmetic, the boy or girl is given a set 
of rules, which present themselves as neither true nor 
false, but as merely the will of the teacher, the way in 
which, for some unfathomable reason, the teacher prefers 
to have the game played. To some degree, in a study of 
such definite practical utility, this is no doubt unavoid 
able ; but as soon as possible, the reasons of rules should 
be set forth by whatever means most readily appeal to 
the childish mind. In geometry, instead of the tedious 
apparatus of fallacious proofs for obvious truisms which 
constitutes the beginning of Euclid, the learner should 
be allowed at first to assume the truth of everything 
obvious, and should be instructed in the demonstrations 
of theorems which are at once startling and easily verifi 
able by actual drawing, such as those in which it is shown 
that three or more lines meet in a point. In this way 
belief is generated ; it is seen that reasoning may lead 
to startling conclusions, which nevertheless the facts will 
verify ; and thus the instinctive distrust of whatever is 
abstract or rational is gradually overcome. Where 
theorems are difficult, they should be first taught as 
exercises in geometrical drawing, until the figure has 
become thoroughly familiar ; it will then be an agreeable 
advance to be taught the logical connections of the 
various lines or circles that occur. It is desirable also 
that the figure illustrating a theorem should be drawn in 
all possible cases and shapes, that so the abstract relations 
with which geometry is concerned may of themselves 



THE STUDY OF MATHEMATICS 63 

emerge as the residue of similarity amid such great 
apparent diversity. In this way the abstract demon 
strations should form but a small part of the instruction, 
and should be given when, by familiarity with concrete 
illustrations, they have come to be felt as *e natural 
embodiment of visible fact. In this early stage proofs 
should not be given with pedantic fullness ; definitely 
fallacious methods, such as that of superposition, should 
be rigidly excluded from the first, but where, without 
such methods, the proof would be very difficult, the 
result should be rendered acceptable by arguments and 
illustrations which are explicitly contrasted with demon 
strations. 

In the beginning of algebra, even the most intelligent 
child finds, as a rule, very great difficulty. The use of 
letters is a mystery, which seems to have no purpose 
except mystification. It is almost impossible, at first, 
not to think that every letter stands for some particular 
number, if only the teacher would reveal what number it 
stands for. The fact is, that in algebra the mind is first 
taught to consider general truths, truths which are not 
asserted to hold only of this or that particular thing, but 
of any one of a whole group of things. It is in the power 
of understanding and discovering such truths that the 
mastery of the intellect over the whole world of things 
actual and possible resides ; and ability to deal with the 
general as such is one of the gifts that a mathematical 
education should bestow. But how little, as a rule, is 
the teacher of algebra able to explain the chasm which 
divides it from arithmetic, and how little is the learner 
assisted in his groping efforts at comprehension ! Usually 
the method that has been adopted in arithmetic is con 
tinued : rules are set forth, with no adequate explanation 
of their grounds ; the pupil learns to use the rules blindly, 



64 MYSTICISM AND LOGIC 

and presently, when he is able to obtain the answer that 
the teacher desires, he feels that he has mastered the 
difficulties of the subject. But of inner comprehension 
of the processes employed he has probably acquired 
almost nothing. 

When algebra has been learnt, all goes smoothly until 
we reach those studies in which the notion of infinity is 
employed the infinitesimal calculus and the whole of 
higher mathematics. The solution of the difficulties 
which formerly surrounded the mathematical infinite is 
probably the greatest achievement of which our own age 
has to boast. Since the beginnings of Greek thought 
these difficulties have been known ; in every age the finest 
intellects have vainly endeavoured to answer the appar 
ently unanswerable questions that had been asked by 
Zeno the Eleatic. At last Georg Cantor has found the 
answer, and has conquered for the intellect a new and 
vast province which had been given over to Chaos and 
old Night. It was assumed as self-evident, until Cantor 
and Dedekind established the opposite, that if, from any 
collection of things, some were taken away, the number 
of things left must always be less than the original 
number of things. This assumption, as a matter of fact, 
holds only of finite collections ; and the rejection of it v 
where the infinite is concerned, has been shown to remove 
all the difficulties that had hitherto baffled human reason 
in this matter, and to render possible the creation of 
an exact science of the infinite. This stupendous fact 
ought to produce a revolution in the higher teaching 
of mathematics ; it has itself added immeasurably to 
the educational value of the subject, and it has at last 
given the means of treating with logical precision many 
studies which, until lately, were wrapped in fallacy 
and obscurity. By those who were educated on the 



THE STUDY OF MATHEMATICS 65 

old lines, the new work is considered to be appallingly 
difficult, abstruse, and obscure ; and it must be con 
fessed that the discoverer, as is so often the case, has 
hardly himself emerged from the mists which the light 
of his intellect is dispelling. But inherently, the new 
doctrine of the infinite, to all candid and inquiring 
minds, has facilitated the mastery of higher mathematics ; 
for hitherto, it has been necessary to learn, by a long 
process of sophistication, to give assent to arguments 
which, on first acquaintance, were rightly judged to be 
confused and erroneous. So far from producing a fear 
less belief in reason, a bold rejection of whatever failed 
to fulfil the strictest requirements of logic, a mathematical 
training, during the past two centuries, encouraged the 
belief that many things, which a rigid inquiry would 
reject as fallacious, must yet be accepted because they 
work in what the mathematician calls " practice." By 
this means, a timid, compromising spirit, or else a sacer 
dotal belief in mysteries not intelligible to the profane, 
has been bred where reason alone should have ruled. All 
this it is now time to sweep away ; let those who wish to 
penetrate into the arcana of mathematics be taught at 
once the true theory in all its logical purity, and in the 
concatenation established by the very essence of the 
entities concerned. 

If we are considering mathematics as an end in itself, 
and not as a technical training for engineers, it is very 
desirable to preserve the purity and strictness of its 
reasoning. Accordingly those who have attained a 
sufficient familiarity with its easier portions should be 
led backward from propositions to which they have 
assented as self-evident to more and more fundamental 
principles from which what had previously appeared as 
premises can be deduced. They should be taught 



66 MYSTICISM AND LOGIC 

what the theory of infinity very aptly illustrates that 
many propositions seem self-evident to the untrained 
mind which, nevertheless, a nearer scrutiny shows to be 
false. By this means they will be led to a sceptical 
inquiry into first principles, an examination of the 
foundations upon which the whole edifice of reasoning is 
built, or, to take perhaps a more fitting metaphor, the 
great trunk from which the spreading branches spring. 
At this stage, it is well to study afresh the elementary 
portions of mathematics, asking no longer merely whether 
a given proposition is true, but also how it grows out of 
the central principles of logic. Questions of this nature 
can now be answered with a precision and certainty 
which were formerly quite impossible ; and in the chains 
of reasoning that the answer requires the unity of all 
mathematical studies at last unfolds itself. 

In the great majority of mathematical text-books there 
is a total lack of unity in method and of systematic 
development of a central theme. Propositions of very 
diverse kinds are proved by whatever means are thought 
most easily intelligible, and much space is devoted to 
mere curiosities which in no way contribute to the main 
argument. But in the greatest works, unity and in 
evitability are felt as in the unfolding of a drama ; in the 
premisses a subject is proposed for consideration, and in 
every subsequent step some definite advance is made 
towards mastery of its nature. The love of system, of 
interconnection, which is perhaps the inmost essence of 
the intellectual impulse, can find free play in mathematics 
as nowhere else. The learner who feels this impulse 
must not be repelled by an array of meaningless examples 
or distracted by amusing oddities, but must be encouraged 
to dwell upon central principles, to become familiar with 
the structure of the various subjects which are put before 



THE STUDY OF MATHEMATICS 67 

him, to travel easily over the steps of the more important 
deductions. In this way a good tone of mind is cultivated, 
and selective attention is taught to dwell by preference 
upon what is weighty and essential. 

When the separate studies into which mathematics is 
divided have each been viewed as a logical whole, as a 
natural growth from the propositions which constitute 
their principles, the learner will be able to understand 
the fundamental science which unifies and systematises 
the whole of deductive reasoning. This is symbolic logic 
a study which, though it owes its inception to Aristotle, 
is yet, in its wider developments, a product, almost 
wholly, of the nineteenth century, and is indeed, in the 
present day, still growing with great rapidity. The true 
method of discovery in symbolic logic, and probably also 
the best method for introducing the study to a learner 
acquainted with other parts of mathematics, is the 
analysis of actual examples of deductive reasoning, with 
a view to the discovery of the principles employed. These 
principles, for the most part, are so embedded in our 
ratiocinative instincts, that they are employed quite un 
consciously, and can be dragged to light only by much 
patient effort. But when at last they have been found, 
they are seen to be few in number, and to be the sole 
source of everything in pure mathematics. The dis 
covery that all mathematics follows inevitably from a 
small collection of fundamental laws is one which im 
measurably enhances the intellectual beauty of the whole ; 
to those who have been oppressed by the fragmentary and 
incomplete nature of most existing chains of deduction 
this discovery comes with all the overwhelming force of a 
revelation ; like a palace emerging from the autumn 
mist as the traveller ascends an Italian hill-side, the 
stately storeys of the mathematical edifice appear in their 



68 MYSTICISM AND LOGIC 

due order and proportion, with a new perfection in every 
part. 

Until symbolic logic had acquired its present develop 
ment, the principles upon which mathematics depends 
were always supposed to be philosophical, and discover 
able only by the uncertain, unprogressive methods 
hitherto employed by philosophers. So long as this was 
thought, mathematics seemed to be not autonomous, but 
dependent upon a study which had quite other methods 
than its own. Moreover, since the nature of the postulates 
from which arithmetic, analysis, and geometry are to be 
deduced was wrapped in all the traditional obscurities of 
metaphysical discussion, the edifice built upon such 
dubious foundations began to be viewed as no better 
than a castle in the air. In this respect, the discovery 
that the true principles are as much a part of mathe 
matics as any of their consequences has very greatly 
increased the intellectual satisfaction to be obtained. 
This satisfaction ought not to be refused to learners 
capable of enjoying it, for it is of a kind to increase our 
respect for human powers and our knowledge of the 
beauties belonging to the abstract world. 

Philosophers have commonly held that the laws of 
logic, which underlie mathematics, are laws of thought, 
laws regulating the operations of our minds. By this 
opinion the true dignity of reason is very greatly lowered : 
it ceases to be an investigation into the very heart ana 
immutable essence of all things actual and possible, be 
coming, instead, an inquiry into something more or less 
human and subject to our limitations. The contemplation 
of what is non-human, the discovery that our minds are 
capable of dealing with material not created by them, 
above all, the realisation that beauty belongs to the outer 
world as to the inner, are the chief means of overcoming 



THE STUDY OF MATHEMATICS 69 

the terrible sense of impotence, of weakness, of exile amid 
hostile powers, which is too apt to result from acknow 
ledging the ail-but omnipotence of alien forces. To 
reconcile us, by the exhibition of its awful beauty, to the 
reign of Fate which is merely the literary personifica 
tion of these forces is the task of tragedy. But mathe 
matics takes us still further from what is human, into the 
region of absolute necessity, to which not only the actual 
world, but every possible world, must conform ; and 
even here it builds a habitation, or rather finds a habita 
tion eternally standing, where our ideals are fully satisfied 
and our best hopes are not thwarted. It is only when we 
thoroughly understand the entire independence of our 
selves, which belongs to this world that reason finds, that 
we can adequately realise the profound importance of its 
beauty. 

Not only is mathematics independent of us and our 
thoughts, but in another sense we and the whole universe 
of existing things are independent of mathematics. The 
apprehension of this purely ideal character is indispens 
able, if we are to understand rightly the place of 
mathematics as one among the arts. It was formerly sup 
posed that pure reason could decide, in some respects, as 
to the nature of the actual world : geometry, at least, was 
thought to deal with the space in which we live. But we 
now know that pure mathematics can never pronounce 
upon questions of actual existence : the world of reason, 
in a sense, controls the world of fact, but it is not at any 
point creative of fact, and in the application of its results 
to the world in time and space, its certainty and precision 
are lost among approximations and working hypotheses. 
The objects considered by mathematicians have, in the 
past, been mainly of a kind suggested by phenomena ; 
but from such restrictions the abstract imagination 



70 MYSTICISM AND LOGIC 

should be wholly free. A reciprocal liberty must thus be 
accorded : reason cannot dictate to the world of facts, 
but the facts cannot restrict reason s privilege of dealing 
with whatever objects its love of beauty may cause to 
seem worthy of consideration. Here, as elsewhere, we 
build up our own ideals out of the fragments to be found 
in the world ; and in the end it is hard to say whether 
the result is a creation or a discovery. 

It is very desirable, in instruction, not merely to per 
suade the student of the accuracy of important theorems, 
but to persuade him in the way which itself has, of all 
possible ways, the most beauty. The true interest of a 
demonstration is not, as traditional modes of exposition 
suggest, concentrated wholly in the result ; where this 
does occur, it must be viewed as a defect, to be remedied, 
if possible, by so generalising the steps of the proof that 
each becomes important in and for itself. An argument 
which serves only to prove a conclusion is like a story 
subordinated to some moral which it is meant to teach : 
for aesthetic perfection no part of the whole should be 
merely a means. A certain practical spirit, a desire for 
rapid progress, for conquest of new realms, is responsible 
for the undue emphasis upon results which prevails in 
mathematical instruction. The better way is to propose 
some theme for consideration in geometry, a figure 
having important properties ; in analysis, a function of 
which the study is illuminating, and so on. Whenever 
proofs depend upon some only of the marks by which we 
define the object to be studied, these marks should be 
isolated and investigated on their own account. For it 
is a defect, in an argument, to employ more premisses 
than the conclusion demands : what mathematicians call 
elegance results from employing only the essential prin 
ciples in virtue of which the thesis is true. It is a merit in 



THE STUDY OF MATHEMATICS 71 

Euclid that he advances as far as he is able to go without 
employing the axiom of parallels not, as is often said, 
because this axiom is inherently objectionable, but 
because, in mathematics, every new axiom diminishes 
the generality of the resulting theorems, and the greatest 
possible generality is before all things to be sought. 

Of the effects of mathematics outside its own sphere 
more has been written than on the subject of its own 
proper ideal. The effect upon philosophy has, in the 
past, been most notable, but most varied ; in the seven 
teenth century, idealism and rationalism, in the eigh 
teenth, materialism and sensationalism, seemed equally 
its offspring. Of the effect which it is likely to have in 
the future it would be very rash to say much ; but in 
one respect a good result appears probable. Against 
that kind of scepticism which abandons the pursuit of 
ideals because the road is arduous and the goal not cer 
tainly attainable, mathematics, within its own sphere, is 
a complete answer. Too often it is said that there is no 
absolute truth, but only opinion and private judgment ; 
that each of us is conditioned, in his view of the world, 
by his own peculiarities, his own taste and bias ; that 
there is no external kingdom of truth to which, by patience 
and discipline, we may at last obtain admittance, but only 
truth for me, for you, for every separate person. By this 
habit of mind one of the chief ends of human effort is 
denied, and the supreme virtue of candour, of fearless 
acknowledgment of what is, disappears from our moral 
vision. Of such scepticism mathematics is a perpetual 
reproof ; ful its edifice of truths stands unshakable and 
inexpugnable to all the weapons of doubting cynicism. 

The effects of mathematics upon practical life, though 
they should not be regarded as the motive of our studies, 
may be used to answer a doubt to which the solitary 



72 MYSTICISM AND LOGIC 

student must always be liable. In a world so full of evil 
and suffering, retirement into the cloister of contempla 
tion, to the enjoyment of delights which, however noble, 
must always be for the few only, cannot but appear as a 
somewhat selfish refusal to share the burden imposed 
upon others by accidents in which justice plays no part. 
Have any of us the right, we ask, to withdraw from 
present evils, to leave our fellow-men unaided, while we 
live a life which, though arduous and austere, is yet 
plainly good in its own nature ? When these questions 
arise, the true answer is, no doubt, that some must keep 
alive the sacred fire, some must preserve, in every genera 
tion, the haunting vision which shadows forth the goal of 
so much striving. But when, as must sometimes occur, 
this answer seems too cold, when we are almost maddened 
by the spectacle of sorrows to which we bring no help, 
then we may reflect that indirectly the mathematician 
often does more for human happiness than any of his 
more practically active contemporaries. The history of 
science abundantly proves that a body of abstract pro 
positions even if, as in the case of conic sections, it 
remains two thousand years without effect upon daily 
life may yet, at any moment, be used to cause a revolu 
tion in the habitual thoughts and occupations of every 
citizen. The use of steam and electricity to take striking 
instances is rendered possible only by mathematics. In 
the results of abstract thought the world possesses a 
capital of which the employment in enriching the common 
round has no hitherto discoverable limits. Nor does 
experience give any means of deciding what parts of 
mathematics will be found useful. Utility, therefore, 
can be only a consolation in moments of discouragement, 
not a guide in directing our studies. 

For the health of the moral life, for ennobling the tone 



THE STUDY OF MATHEMATICS 73 

of an age or a nation, the austerer virtues have a strange 
power, exceeding the power of those not informed and 
purified by thought. Of these austerer virtues the love of 
truth is the chief, and in mathematics, more than else 
where, the love of truth may find encouragement for wan 
ing faith. Every great study is not only an end in itself, but 
also a means of creating and sustaining a lofty habit of 
mind ; and this purpose should be kept always in view 
throughout the teaching and learning of mathematics. 



V 

MATHEMATICS AND THE 
METAPHYSICIANS 

THE nineteenth century, which prided itself upon 
the invention of steam and evolution, might have 
derived a more legitimate title to fame from the discovery 
of pure mathematics. This science, like most others, 
was baptised long before it was born ; and thus we find 
writers before the nineteenth century alluding to what 
they called pure mathematics. But if they had been 
asked what this subject was, they would only have been 
able to say that it consisted of Arithmetic, Algebra, 
Geometry, and so on. As to what these studies had in 
common, and as to what distinguished them from applied 
mathematics, our ancestors were completely in the dark. 
Pure mathematics was discovered by Boole, in a work 
which he called the Laws of Thought (1854). This work 
abounds in asseverations that it is not mathematical, 
the fact being that Boole was too modest to suppose his 
book the first ever written on mathematics. He was also 
mistaken in supposing that he was dealing with the laws 
of thought : the question how people actually think was 
quite irrelevant to him, and if his book had really con 
tained the laws of thought, it was curious that no one 
should ever have thought in such a way before. His 
book was in fact concerned with formal logic, and this 
IB the same thing as mathematics 

74 



MATHEMATICS AND METAPHYSICIANS 75 

Pure mathematics consists entirely of assertions to the 
effect that, if such and such a proposition is true of any 
thing, then such and such another proposition is true of 
that thing. It is essential not to discuss whether the first 
proposition is really true, and not to mention what the 
anything is, of which it is supposed to be true. Both 
these points would belong to applied mathematics. We 
start, in pure mathematics, from certain rules of infer 
ence, by which we can infer that if one proposition is 
true, then so is some other proposition. These rules of 
inference constitute the major part of the principles of 
formal logic. We then take any hypothesis that seems 
amusing, and deduce its consequences, //our hypothesis 
is about anything, and not about some oneormore particular 
things, then our deductions constitute mathematics. Thus 
mathematics may be defined as the subject in which we 
never know what we are talking about, nor whether what 
we are saying is true. People who have been puzzled by the 
beginnings of mathematics will, I hope, find comfort in 
this definition, and will probably agree that it is accurate. 

As one of the chief triumphs of modern mathematics 
consists in having discovered what mathematics really 
is, a few more words on this subject may not be amiss. 
It is common to start any branch of mathematics for 
instance, Geometry with a certain number of primitive 
ideas, supposed incapable of definition, and a certain 
number of primitive propositions or axioms, supposed 
incapable of proof. Now the fact is that, though there 
are indefinables and indemonstrables in every branch of 
applied mathematics, there are none in pure mathematics 
except such as belong to general logic. Logic, broadly 
speaking, is distinguished by the fact that its propositions 
can be put into a form in which they apply to anything 
whatever. All pure mathematics Arithmetic, Analysis, 



76 MYSTICISM AND LOGIC 

and Geometry is built up by combinations of the primi 
tive ideas of logic, and its propositions are deduced from 
the general axioms of logic, such as the syllogism and the 
other rules of inference. And this is no longer a dream 
or an aspiration. On the contrary, over the greater and 
more difficult part of the domain of mathematics, it has 
been already accomplished ; in the few remaining cases, 
there is no special difficulty, and it is now being rapidly 
achieved. Philosophers have disputed for ages whether 
such deduction was possible ; mathematicians have sat 
down and made the deduction. For the philosophers 
there is now nothing left but graceful acknowledg 
ments. 

The subject of formal logic, which has thus at last 
shown itself to be identical with mathematics, was, as 
every one knows, invented by Aristotle, and formed the 
chief study (other than theology) of the Middle Ages. 
But Aristotle never got beyond the syllogism, which is a 
very small part of the subject, and the schoolmen never 
got beyond Aristotle. If any proof were required of our 
superiority to the mediaeval doctors, it might be found in 
this. Throughout the Middle Ages, almost all the best 
intellects devoted themselves to formal logic, whereas in 
the nineteenth century only an infinitesimal proportion of 
the world s thought went into this subject. Nevertheless, 
in each decade since 1850 more has been done to advance 
the subject than in the whole period from Aristotle to 
Leibniz. People have discovered how to make reasoning 
symbolic, as it is in Algebra, so that deductions are 
effected by mathematical rules. They have discovered 
many rules besides the syllogism, and a new branch of 
logic, called the Logic of Relatives, 1 has been invented 
to deal with topics that wholly surpassed the powers of 
1 This subject is due in the main to Mr. C. S. Peirce. 



MATHEMATICS AND METAPHYSICIANS 77 

the old logic, though they form the chief contents of 
mathematics. 

It is not easy for the lay mind to realise the importance 
of symbolism in discussing the foundations of mathe 
matics, and the explanation may perhaps seem strangely 
paradoxical. The fact is that symbolism is useful because 
it makes things difficult. (This is not true of the advanced 
parts of mathematics, but only of the beginnings.) What 
we wish to know is, what can be deduced from what. 
Now, in the beginnings, everything is self-evident ; and 
it is very hard to see whether one self-evident proposition 
follows from another or not. Obviousness is always the 
enemy to correctness. Hence we invent some new and 
difficult symbolism, in which nothing seems obvious. 
Then we set up certain rules for operating on the symbols, 
and the whole thing becomes mechanical. In this way 
we find out what must be taken as premiss and what can 
be demonstrated or defined. For instance, the whole of 
Arithmetic and Algebra has been shown to require three 
indefinable notions and five indemonstrable propositions. 
But without a symbolism it would have been very hard 
to find this out. It is so obvious that two and two are four, 
that we can hardly make ourselves sufficiently sceptical 
to doubt whether it can be proved. And the same holds 
in other cases where self-evident things are to be proved. 

But the proof of self-evident propositions may seem, to 
the uninitiated, a somewhat frivolous occupation, To 
this we might reply that it is often by no means self- 
evident that one obvious proposition follows from another 
obvious proposition ; so that we are really discovering 
new truths when we prove what is evident by a method 
which is not evident. But a more interesting retort is, 
that since people have tried to prove obvious propositions, 
they have found that many of them are false. Self- 



78 MYSTICISM AND LOGIC 

evidence is often a mere will-o -the-wisp, which is sure to 
lead us astray if we take it as our guide. For instance, 
nothing is plainer than that a whole always has more 
terms than a part, or that a number is increased by add 
ing one to it. But these propositions are now known to 
be usually false. Most numbers are infinite, and if a 
number is infinite you may add ones to it as long as you 
like without disturbing it in the least. One of the merits 
of a proof is that it instils a certain doubt as to the result 
proved ; and when what is obvious can be proved in 
some cases, but not in others, it becomes possible to sup 
pose that in these other cases it is false. 

The great master of the art of formal reasoning, among 
the men of our own day, is an Italian, Professor Peano, 
of the University of Turin. 1 He has reduced the greater 
part of mathematics (and he or his followers will, in time, 
have reduced the whole) to strict symbolic form, in which 
there are no words at all. In the ordinary mathematical 
books, there are no doubt fewer words than most readers 
would wish. Still, little phrases occur, such as therefore, 
let us assume, consider, or hence it follows. All these, how 
ever, are a concession, and are swept away by Professor 
Peano. For instance, if we wish to learn the whole of 
Arithmetic, Algebra, the Calculus, and indeed all that is 
usually called pure mathematics (except Geometry), we 
must start with a dictionary of three words. One symbol 
stands for zero, another for number, and a third for next 
after. What these ideas mean, it is necessary to know if 
you wish to become an arithmetician. But after symbols 
have been invented for these three ideas, not another 
word is required in the whole development. All future 
symbols are symbolically explained by means of these 

1 I ought to have added Frege, but his writings were unknown to 
me when this article was written. [Note added in 1917.] 



MATHEMATICS AND METAPHYSICIANS 79 

three. Even these three can be explained by means of 
the notions of relation and class ; but this requires the 
Logic of Relations, which Professor Peano has never 
taken up. It must be admitted that what a mathe 
matician has to know to begin with is not much. There 
are at most a dozen notions out of which all the notions 
in all pure mathematics (including Geometry) are com 
pounded. Professor Peano, who is assisted by a very 
able school of young Italian disciples, has shown how 
this may be done ; and although the method which he 
has invented is capable of being carried a good deal 
further than he has carried it, the honour of the pioneer 
must belong to him. 

Two hundred years ago, Leibniz foresaw the science 
which Peano has perfected, and endeavoured to create it 
He was prevented from succeeding by respect for the 
authority of Aristotle, whom he could not believe guilty 
of definite, formal fallacies ; but the subject which he 
desired to create now exists, in spite of the patronising 
contempt with which his schemes have been treated by all 
superior persons. From this " Universal Characteristic," 
as he called it, he hoped for a solution of all problems, 
and an end to all disputes. " If controversies were to 
arise," he says, " there would be no more need of dis 
putation between two philosophers than between two 
accountants. For it would suffice to take their pens in 
their hands, to sit down to their desks, and to say to 
each other (with a friend as witness, if they liked), Let 
us calculate/ This optimism has now appeared to be 
somewhat excessive ; there still are problems whose 
solution is doubtful, and disputes which calculation 
cannot decide. But over an enormous field of what was 
formerly controversial, Leibniz s dream has become sober 
fact. In the whole philosophy of mathematics, which 



8o MYSTICISM AND LOGIC 

used to be at least as full of doubt as any other part of 
philosophy, order and certainty have replaced the con 
fusion and hesitation which formerly reigned. Philo 
sophers, of course, have not yet discovered this fact, and 
continue to write on such subjects in the old way. But 
mathematicians, at least in Italy, have now the power ol 
treating the principles of mathematics in an exact and 
masterly manner, by means of which the certainty of 
mathematics extends also to mathematical philosophy. 
Hence many of the topics which used to be placed among 
the great mysteries for example, the natures of infinity, 
of continuity, of space, time and motion are now no 
longer in any degree open to doubt or discussion. Those 
who wish to know the nature of these things need only 
read the works of such men as Peano or Georg Cantor ; 
they will there find exact and indubitable expositions of 
all these quondam mysteries. 

In this capricious world, nothing is more capricious 
than posthumous fame. One of the most notable examples 
of posterity s lack of judgment is the Eleatic Zeno. This 
man, who may be regarded as the founder of the philo 
sophy of infinity, appears in Plato s Parmenides in the 
privileged position of instructor to Socrates. He invented 
four arguments, all immeasurably subtle and profound, 
to prove that motion is impossible, that Achilles can 
never overtake the tortoise, and that an arrow in flight 
is really at rest. After being refuted by Aristotle, and 
by every subsequent philosopher from that day to our 
own, these arguments were reinstated, and made the 
basis of a mathematical renaissance, by a German pro 
fessor, who probably never dreamed of any connection 
between himself and Zeno. Weierstrass, l by strictly 

1 Professor of Mathematics in the University of Berlin. He died in 
1897. 



MATHEMATICS AND METAPHYSICIANS 81 

banishing from mathematics the use of infinitesimals, 
has at last shown that we live in an unchanging world, 
and that the arrow in its flight is truly at rest. 
Zeno s only error lay in inferring (if he did infer) 
that, because there is no such thing as a state of 
change, therefore the world is in the same state 
at any one time as at any other. This is a conse 
quence which by no means follows ; and in this respect, 
the German mathematician is more constructive than 
the ingenious Greek. Weierstrass has been able, by 
embodying his views in mathematics, where familiarity 
with truth eliminates the vulgar prejudices of common 
sense, to invest Zeno s paradoxes with the respectable 
air of platitudes ; and if the result is less delightful to the 
lover of reason than Zeno s bold defiance, it is at any 
rate more calculated to appease the mass of academic 
mankind. 

Zeno was concerned, as a matter of fact, with three 
problems, each presented by motion, but each more 
abstract than motion, and capable of a purely arith 
metical treatment. These are the problems of the 
infinitesimal, the infinite, and continuity. To state 
clearly the difficulties involved, was to accomplish perhaps 
the hardest part of the philosopher s task. This was done 
by Zeno. From him to our own day, the finest intellects 
of each generation in turn attacked the problems, but 
achieved, broadly speaking, nothing. In our own time, 
however, three men Weierstrass, Dedekind, and Cantor 
have not merely advanced the three problems, but have 
completely solved them. The solutions, for those ac 
quainted with mathematics, are so clear as to leave no 
longer the slightest doubt or difficulty. This achieve 
ment is probably the greatest of which our age has to 
boast ; and I know of no age (except perhaps the golden 



82 MYSTICISM AND LOGIC 

age oi Sreece) which has a more convincing proof to offer 
of the transcendent genius of its great men. Of the three 
problems, that of the infinitesimal was solved by Weier- 
strass ; the solution of the other two was begun by 
Dedekind, and definitively accomplished by Cantor. 

The infinitesimal played formerly a great part in 
mathematics. It was introduced by the Greeks, who 
regarded a circle as differing infinitesimally from a polygon 
with a very large number of very small equal sides. It 
gradually grew in importance, until, when Leibniz in 
vented the Infinitesimal Calculus, it seemed to become 
the fundamental notion of all higher mathematics. 
Carlyle tells, in his Frederick the Great, how Leibniz used 
to discourse to Queen Sophia Charlotte of Prussia con 
cerning the infinitely little, and how she would reply that 
on that subject she needed no instruction the behaviour 
of courtiers had made her thoroughly familiar with it. 
But philosophers and mathematicians who for the most 
part had less acquaintance with courts continued to 
discuss this topic, though without making any advance. 
The Calculus required continuity, and continuity was 
supposed to require the infinitely little ; but nobody 
could discover what the infinitely little might be. It was 
plainly not quite zero, because a sufficiently large number 
of infinitesimals, added together, were seen to make up a 
finite whole. But nobody could point out any fraction 
which was not zero, and yet not finite. Thus there was a 
deadlock. But at last Weierstrass discovered that the 
infinitesimal was not needed at all, and that everything 
could be accomplished without it. Thus there was no 
longer any need to suppose that there was such a thing. 
Nowadays, therefore, mathematicians are more dignified 
than Leibniz : instead of talking about the infinitely 
small, they talk about the infinitely great a subject 



MATHEMATICS AND METAPHYSICIANS 83 

which, however appropriate to monarchs, seems, un 
fortunately, to interest them even less than the infinitely 
li^le interested the monarchs to whom Leibniz discoursed. 

The banishment of the infinitesimal has all sorts of odd 
consequences, to which one has to become gradually 
accustomed. For example, there is no such thing as the 
next moment. The interval between one moment and the 
next would have to be infinitesimal, since, if we take two 
moments with a finite interval between them, there are 
always other moments in the interval. Thus if there are 
to be no infinitesimals, no two moments are quite con 
secutive, but there are always other moments between any 
two. Hence there must be an infinite number of moments 
between any two ; because if there were a finite number 
one would be nearest the first of the two moments, and 
therefore next to it. This might be thought to be a difn 
culty ; but, as a matter of fact, it is here that the philo 
sophy of the infinite comes in, and makes all straight. 

The same sort of thing happens in space. If any piece 
of matter be cut in two, and then each part be halved, 
and so on, the bits will become smaller and smaller, and 
can theoretically be made as small as we please. However 
small they may be, they can still be cut up and made 
smaller still. But they will always have some finite size, 
however small they may be. We never reach the in 
finitesimal in this way, and no finite number of divisions 
will bring us to points. Nevertheless there are points, 
only these are not to be reached by successive divisions. 
Here again, the philosophy of the infinite shows us how 
this is possible, and why points are not infinitesimal 
lengths. .... 

As regards motion and change, we get similarly curious 
results. People used to think that when a thing changes, 
it must be in a state of change, and that when a thing 



84 MYSTICISM AND LOGIC 

moves, it is in a state of motion. This is now known to 
be a mistake. When a body moves, all that can be said 
is that it is in one place at one time and in another at 
another. We must not say that it will be in a neighbour 
ing place at the next instant, since there is no next 
instant. Philosophers often tell us that when a body is 
in motion, it changes its position within the instant. To 
this view Zeno long ago made the fatal retort that every 
body always is where it is$ but a retort so simple and 
brief was not of the kind to which philosophers are accus 
tomed to give weight, and they have continued down to 
our own day to repeat the same phrases which roused the 
Eleatic s destructive ardour. It was only recently that 
it became possible to explain motion in detail in accord 
ance with Zeno s platitude, and in opposition to the 
philosopher s paradox. We may now at last indulge the 
comfortable belief that a body in motion is just as truly 
where it is as a body at rest. Motion consists merely in 
the fact that bodies are sometimes in one place and some 
times in another, and that they are at intermediate places 
at intermediate times. Only those who have waded 
through the quagmire of philosophic speculation on this 
subject can realise what a liberation from antique pre 
judices is involved in this simple and straightforward 
commonplace. 

The philosophy of the infinitesimal, as we have just 
seen, is mainly negative. People used to believe in it, 
and now they have found out their mistake. The philo 
sophy of the infinite, on the other hand, is wholly positive. 
It was formerly supposed that infinite numbers, and the 
mathematical infinite generally, were self-contradictory. 
But as it was obvious that there were infinities for 
example, the number of numbers the contradictions of 
infinity seemed unavoidable, and philosophy seemed to 



MATHEMATICS AND METAPHYSICIANS 85 \ 

A 
have wandered into a " cul-de-sac." This difficulty led ^ 

to Kant s antinomies, and hence, more or less indirectly, 
to much of Hegel s dialectic method. Almost all current 
philosophy is upset by the fact (of which very few philo 
sophers are as yet aware) that all the ancient and respect 
able contradictions in the notion of the infinite have been 
once for all disposed of. The method by which this has 
been done is most interesting and instructive. In the 
first place, though people had talked glibly about infinity 
ever since the beginnings of Greek thought, nobody had 
ever thought of asking, What is infinity ? If any 
philosopher had been asked for a definition of infinity, he 
might have produced some unintelligible rigmarole, but he 
would certainly not have been able to give a definition 
that had any meaning at all. Twenty years ago, roughly 
speaking, Dedekind and Cantor asked this question, and, 
what is more remarkable, they answered it. They found, 
that is to say, a perfectly precise definition of an infinite 
number or an infinite collection of things. This was the 
first and perhaps the greatest step. It then remained to 
examine the supposed contradictions in this notion. 
Here Cantor proceeded in the only proper way. He took 
pairs of contradictory propositions, in which both sides 
of the contradiction would be usually regarded as demon 
strable, and he strictly examined the supposed proofs. He 
found that all proofs adverse to infinity involved a certain 
principle, at first sight obviously true, but destructive, 
in its consequences, of almost all mathematics. The 
proofs favourable to infinity, on the other hand, involved 
no principle that had evil consequences. It thus appeared 
that common sense had allowed itself to be taken in by a 
specious maxim, and that, when once this maxim was 
rejected, all went well. 
The maxim in question is, that if one collection is part 



86 MYSTICISM AND LOGIC 

of another, the one which is a part has fewer terms than 
the one of which it is a part. This maxim is true of finite 
numbers. For example, Englishmen are only some among 
Europeans, and there are fewer Englishmen than Euro 
peans. But when we come to infinite numbers, this is no 
longer true. This breakdown of the maxim gives us the 
precise definition of infinity. A collection of terms is 
infinite when it contains as parts other collections which 
have just as many terms as it has. If you can take away 
some of the terms of a collection, without diminishing 
the number of terms, then there are an infinite number 
of terms in the collection. For example, there are just 
as many even numbers as there are numbers altogether, 
since every number can be doubled. This may be seen 
by putting odd and even numbers together in one row, 
and even numbers alone in a row below : 

1, 2, 3, 4, 5, ad infinitum. 

2, 4, 6, 8, 10, ad infinitum. 

There are obviously just as many numbers in the row 
below as in the row above, because there is one below for 
each one above. This property, which was formerly 
thought to be a contradiction, is now transformed into a 
harmless definition of infinity, and shows, in the above 
case, that the number of finite numbers is infinite. 

But the uninitiated may wonder how it is possible to 
deal with a number which cannot be counted. It is im 
possible to count up all the numbers, one by one, because, 
however many we may count, there are always more to 
follow. The fact is that counting is a very vulgar and 
elementary way of finding out how many terms there 
are in a collection. And in any case, counting gives us 
what mathematicians call the ordinal number of our 
terms ; that is to say, it arranges our terms in an order or 



MATHEMATICS AND METAPHYSICIANS 87 

series, and its result tells us what type of series results 
from this arrangement. In other words, it is impossible 
to count things without counting some first and others 
afterwards, so that counting always has to do with order. 
Now when there are only a finite number of terms, we 
can count them in any order we like ; but when there are 
an infinite number, what corresponds to counting will 
give us quite different results according to the way in 
which we carry out the operation. Thus the ordinal 
number, which results from what, in a general sense 
may be called counting, depends not only upon how many 
terms we have, but also (where the number of terms is 
infinite) upon the way in which the terms are arranged. 

The fundamental infinite numbers are not ordinal, but 
are what is called cardinal. They are not obtained by 
putting our terms in order and counting them, but by a 
different method, which tells us, to begin with, whether two 
collections have the same number of terms, or, if not, 
which is the greater. 1 It does not tell us, in the way in 
which counting does, what number of terms a collection 
has ; but if we define a number as the number of terms 
in such and such a collection, then this method enables 
us to discover whether some other collection that may be 
mentioned has more or fewer terms. An illustration will 
show how this is done. If there existed some country in 
which, for one reason or another, it was impossible to 
take a census, but in which it was known that every man 
had a wife and every woman a husband, then (provided 
polygamy was not a national institution) we should know, 
without counting, that there were exactly as many men 
as there were women in that country, neither more nor 

1 [Note added in 1917.] Although some infinite numbers are 
greater than some others, it cannot be proved that of any two infinite 
numbers one must be the greater. 



88 MYSTICISM AND LOGIC 

less. This method can be applied generally. If there is 
some relation which, like marriage, connects the things 
in one collection each with one of the things in another 
collection, and vice versa, then the two collections have 
the same number of terms. This was the way in which 
we found that there are as many even numbers as there 
are numbers. Every number can be doubled, and every 
even number can be halved, and each process gives just 
one number corresponding to the one that is doubled or 
halved. And in this way we can find any number of 
collections each of which has just as many terms as there 
are finite numbers. If every term of a collection can be 
hooked on to a number, and all the finite numbers are 
used once, and only once, in the process, then our 
collection must have just as many terms as there are 
finite numbers. This is the general method by which the 
numbers of infinite collections are defined. 

But it must not be supposed that all infinite numbers 
are equal. On the contrary, there are infinitely more 
infinite numbers than finite ones. There are more ways 
of arranging the finite numbers in different types of 
series than there are finite numbers. There are probably 
more points in space and more moments in time than 
there are finite numbers. There are exactly as many 
fractions as whole numbers, although there are an infinite 
number of fractions between any two whole numbers. 
But there are more irrational numbers than there are 
whole numbers or fractions. There are probably exactly 
as many points in space as there are irrational numbers, 
and exactly as many points on a line a millionth of an 
inch long as in the whole of infinite space, There is a 
greatest of all infinite numbers, which is the number of 
things altogether, of every sort and kind. It is obvious 
that there cannot be a greater number than this, because, 



MATHEMATICS AND METAPHYSICIANS 89 

if everything has been taken, there is nothing left to add. 
Cantor has a proof that there is no greatest number, and 
if this proof were valid, the contradictions of infinity 
would reappear in a sublimated form. But in this one 
point, the master has been guilty of a very subtle fallacy, 
which I hope to explain in some future work. 1 

We can now understand why Zeno believed that Achilles 
cannot overtake the tortoise and why as a matter of fact 
he can overtake it. We shall see that all the people who 
disagreed with Zeno had no right to do so, because they 
all accepted premises from which his conclusion followed. 
The argument is this : Let Achilles and the tortoise start 
along a road at the same time, the tortoise (as is only 
fair) being allowed a handicap. Let Achilles go twice as 
fast as the tortoise, or ten times or a hundred times as 
fast. Then he will never reach the tortoise. For at every 
moment the tortoise is somewhere and Achilles is some 
where ; and neither is ever twice in the same place while 
the race is going on. Thus the tortoise goes to just as 
many places as Achilles does, because each is in one place 
at one moment, and in another at any other moment. 
But if Achilles were to catch up with the tortoise, the 
places where the tortoise would have been would be only 
part of the places where Achilles would have been. Here, 
we must suppose, Zeno appealed to the maxim that the 
whole has more terms that the part. 2 Thus if Achilles were 

1 Cantor was not guilty of a fallacy on this point. His proof 
that there is no greatest number is valid. The solution of the puzzle 
is complicated and depends upon the theory of types, which is explained 
in Principia Mathematica, Vol. I (Camb. Univ. Press, 1910). [Note 
added in 1917.] 

1 This must not be regarded as a historically correct account of 
what Zeno actually had in mind. It is a new argument for his con 
clusion, not the argument which influenced him. On this point, see 
e.g. C. D. Broad, "Note on Achilles and the Tortoise," Mind, N.S., 
Vol. XXII, pp. 318-19. Much valuable work on the interpretation of 
Zeno has been done sine?- this article was written. [Note added in 1917.] 



90 MYSTICISM AND LOGIC 

to overtake the tortoise, he would have been in more 
places than the tortoise ; but we saw that he must, in any 
period, be in exactly as many places as the tortoise. 
Hence we infer that he can never catch the tortoise. This 
argument is strictly correct, if we allow the axiom that 
the whole has more terms than the part. As the con 
clusion is absurd, the axiom must be rejected, and then 
all goes well. But there is no good word to be said for 
the philosophers of the past two thousand years and 
more, who have all allowed the axiom and denied the 
conclusion. 

The retention of this axiom leads to absolute contra 
dictions, while its rejection leads only to oddities. Some 
of these oddities, it must be confessed, are very odd. 
One of them, which I call the paradox of Tristram Shandy, 
is the converse of the Achilles, and shows that the tortoise, 
if you give him time, will go just as far as Achilles. 
Tristram Shandy, as we know, employed two years in 
chronicling the first two days of his life, and lamented 
that, at this rate, material would accumulate faster than 
he could deal with it, so that, as years went by, he would 
be farther and farther from the end of his history. Now 
I maintain that, if he had lived for ever, and had not 
wearied of his task, then, even if his life had continued 
as eventfully as it began, no part of his biography would 
have remained unwritten. For consider : the hundredth 
day will be described in the hundredth year, the thousandth 
in the thousandth year, and so on. Whatever day we 
may choose as so far on that he cannot hope to reach it, 
that day will be described in the corresponding year. 
Thus any day that may be mentioned will be written up 
sooner or later, and therefore no part of the biography 
will remain permanently unwritten. This paradoxical 
but perfectly true proposition depends upon the fact 



MATHEMATICS AND METAPHYSICIANS 91 

that the number of days in all time is no greater than the 
number of years. 

Thus on the subject of infinity it is impossible to avoid 
conclusions which at first sight appear paradoxical, and 
this is the reason why so many philosophers have supposed 
that there were inherent contradictions in the infinite. 
But a little practice enables one to grasp the true prin 
ciples oi Cantor s doctrine, and to acquire new and 
better instincts as to the true and the false. The oddities 
then become no odder than the people at the antipodes, 
who used to be thought impossible because they would 
find it so inconvenient to stand on their heads. 

The solution of the problems concerning infinity has 
enabled Cantor to solve also the problems of continuity. 
Of this, as of infinity, he has given a perfectly precise 
definition, and has shown that there are no contradictions 
in the notion so defined. But this subject is so technical 
that it is impossible to give any account of it here. 

The notion of continuity depends upon that of order, 
since continuity is merely a particular type of order. 
Mathematics has, in modern times, brought order into 
greater and greater prominence. In former days, it was 
supposed (and philosophers are still apt to suppose) that 
quantity was the fundamental notion of mathematics. 
But nowadays, quantity is banished altogether, except 
from one little corner of Geometry, while order more and 
more reigns supreme. The investigation of different 
kinds of series and their relations is now a very large part 
of mathematics, and it has been found that this investiga 
tion can be conducted without any reference to quantity, 
and, for the most part, without any reference to number. 
All types of series are capable of formal definition, and 
their properties can be deduced from the principles of 
symbolic logic by means of the Algebra of Relatives. 



92 MYSTICISM AND LOGIC 

The notion of a limit, which is fundamental in the greater 
part of higher mathematics, used to be defined by means 
of quantity, as a term to which the terms of some series 
approximate as nearly as we please. But nowadays the 
limit is denned quite differently, and the series which it 
limits may not approximate to it at all. This improve 
ment also is due to Cantor, and it is one which has 
revolutionised mathematics. Only order is now relevant 
to limits. Thus, for instance, the smallest of the infinite 
integers is the limit of the finite integers, though all 
finite integers are at an infinite distance from it. The 
study of different types of series is a general subject of 
which the study of ordinal numbers (mentioned above) is 
a special and very interesting branch. But the unavoid 
able technicalities of this subject render it impossible to 
explain to any but professed mathematicians. 

Geometry, like Arithmetic, has been subsumed, in 
recent times, under the general study of order. It was 
formerly supposed that Geometry was the study of the 
nature of the space in which we live, and accordingly it 
was urged, by those who held that what exists can only 
be known empirically, that Geometry should really be 
regarded as belonging to applied mathematics. But it 
has gradually appeared, by the increase of non-Euclidean 
systems, that Geometry throws no more light upon the 
nature of space than Arithmetic throws upon the popula 
tion of the United States. Geometry is a whole collection 
of deductive sciences based on a corresponding collection 
of sets of axioms. One set of axioms is Euclid s ; other 
equally good sets of axioms lead to other results. Whether 
Euclid s axioms are true, is a question as to which the 
the pure mathematician is indifferent ; and, what is more, 
it is a question which it is theoretically impossible to 
answer with certainty in the affirmative. It might pos- 



MATHEMATICS AND METAPHYSICIANS 93 

sibly be shown, by very careful measurements, that 
Euclid s axioms are false ; but no measurements could 
ever assure us (owing to the errors of observation) that 
they are exactly true. Thus the geometer leaves to the 
man of science to decide, as best he may, what axioms are 
most nearly true in the actual world. The geometer 
takes any set of axioms that seem interesting, and 
deduces their consequences. What defines Geometry, 
in this sense, is that the axioms must give rise to a series 
of more than one dimension . And it is thus that Geometry 
becomes a department in the study of order. 

In Geometry, as in other parts of mathematics, Peano 
and his disciples have done work of the very greatest 
merit as regards principles. Formerly, it was held by 
philosophers and mathematicians alike that the proofs in 
Geometry depended on the figure ; nowadays, this is 
known to be false. In the best books there are no figures 
at all. The reasoning proceeds by the strict rules of 
formal logic from a set of axioms laid down to begin with. 
If a figure is used, all sorts of things seem obviously to 
follow, which no formal reasoning can prove from the 
explicit axioms, and which, as a matter of fact, are only 
accepted because they are obvious. By banishing the 
figure, it becomes possible to discover all the axioms that 
are needed ; and in this way all sorts of possibilities, 
which would have otherwise remained undetected, are 
brought to light. 

One great advance, from the point of view of correct 
ness, has been made by introducing points as they are 
required, and not starting, as was formerly done, by 
assuming the whole of space. This method is due partly 
to Peano, partly to another Italian named Fano. To 
those unaccustomed to it, it has an air of somewhat 
wilful pedantry. In this way, we begin with the following 



94 MYSTICISM AND LOGIC 

axioms : (i) There is a class of entities called points. 
(2) There is at least one point. (3) If a be a point, there 
is at least one other point besides a. Then we bring in 
the straight line joining two points, and begin again with 
(4), namely, on the straight line joining a and b, there is 
at least one other point besides a and b. (5) There is at 
least one point not on the line ab. And so we go on, till 
we have the means of obtaining as many points as we 
require. But the word space, as Peano humorously 
remarks, is one for which Geometry has no use at all. 

The rigid methods employed by modern geometers 
have deposed Euclid from his pinnacle of correctness. It 
was thought, until recent times, that, as Sir Henry Savile 
remarked in 1621, there were only two blemishes in 
Euclid, the theory of parallels and the theory of pro 
portion. It is now known that these are almost the only 
points in which Euclid is free from blemish. Countless 
errors are involved in his first eight propositions. That 
is to say, not only is it doubtful whether his axioms are 
true, which is a comparatively trivial matter, but it is 
certain that his propositions do not follow from the 
axioms which he enunciates. A vastly greater number 
of axioms, which Euclid unconsciously employs, are re 
quired for the proof of his propositions. Even in the 
first proposition of all, where he constructs an equilateral 
triangle on a given base, he uses two circles which are 
assumed to intersect. But no explicit axiom assures us 
that they do so, and in some kinds of spaces they do not 
always intersect. It is quite doubtful whether our space 
belongs to one of these kinds or not. Thus Euclid fails 
entirely to prove his point in the very first proposition. 
As he is certainly not an easy author, and is terribly long- 
winded, he has no longer any but an historical interest. 
Under these circumstances, it is nothing less than a 



MATHEMATICS AND METAPHYSICIANS 95 

scandal that he should still be taught to boys in England. 1 
A book should have either intelligibility or correctness ; 
to combine the two is impossible, but to lack both is to 
be unworthy of such a place as Euclid has occupied in 
education. 

The most remarkable result of modern methods in 
mathematics is the importance of symbolic logic and of 
rigid formalism. Mathematicians, under the influence of 
Weierstrass, have shown in modern times a care for 
accuracy, and an aversion to slipshod reasoning, such as 
had not been known among them previously since the time 
of the Greeks. The great inventions of the seventeenth 
century Analytical Geometry and the Infinitesimal 
Calculus were so fruitful in new results that mathe 
maticians had neither time nor inclination to examine 
their foundations. Philosophers, who should have taken 
up the task, had too little mathematical ability to invent 
the new branches of mathematics which have now been 
found necessary for any adequate discussion. Thus 
mathematicians were only awakened from their " dog 
matic slumbers " when Weierstrass and his followers 
showed that many of their most cherished propositions 
are in general false. Macaulay, contrasting the certainty 
of mathematics with the uncertainty of philosophy, asks 
who ever heard of a reaction against Taylor s theorem ? 
If he had lived now, he himself might have heard of such 
a reaction, for this is precisely one of the theorems which 
modern investigations have overthrown. Such rude 
shocks to mathematical faith have produced that love of 
formalism which appears, to those who are ignorant of 
its motive, to be mere outrageous pedantry. 

1 Since the above was written, he has ceased to be used as a text 
book But I fear many of the books now used are so bad that the 
change is no great improvement. [Note added in 1917.] 



96 MYSTICISM AND LOGIC 

The proof that all pure mathematics, including 
Geometry, is nothing but formal logic, is a fatal blow to 
the Kantian philosophy. Kant, rightly perceiving that 
Euclid s propositions could not be deduced from Euclid s 
axioms without the help of the figures, invented a theory 
of knowledge to account for this fact ; and it accounted 
so successfully that, when the fact is shown to be a mere 
defect in Euclid, and not a result of the nature of geo 
metrical reasoning, Kant s theory also has to be aban 
doned. The whole doctrine of a priori intuitions, by which 
Kant explained the possibility of pure mathematics, is 
wholly inapplicable to mathematics in its present form. 
The Aristotelian doctrines of the schoolmen come nearer 
in spirit to the doctrines which modern mathematics 
inspire ; but the schoolmen were hampered by the fact 
that their formal logic was very defective, and that the 
philosophical logic based upon the syllogism showed a 
corresponding narrowness. What is now required is to 
give the greatest possible development to mathematical 
logic, to allow to the full the importance of relations, and 
then to found upon this secure basis a new philosophical 
logic, which may hope to borrow some of the exactitude 
and certainty of its mathematical foundation. If this 
can be successfully accomplished, there is every reason 
to hope that the near future will be as great an epoch in 
pure philosophy as the immediate past has been in the 
principles of mathematics. Great triumphs inspire great 
hopes ; and pure thought may achieve, within our 
generation, such results as will place our time, in this 
respect, on a level with the greatest age of Greece. 1 

1 The greatest age of Greece was brought to an end by the 
Peloponnesian War. [Note added in 1917.] 



ON SCIENTIFIC METHOD IN 
PHILOSOPHY 

WHEN we try to ascertain the motives which have 
led men to the investigation of philosophical 
questions, we find that, broadly speaking, they can be 
divided into two groups, often antagonistic, and leading 
to very divergent systems. These two groups of motives 
are, on the one hand, those derived from religion and 
ethics, and, on the other hand, those derived from science. 
Plato, Spinoza, and Hegel may be taken as typical of the 
philosophers whose interests are mainly religious and 
ethical, while Leibniz, Locke, and Hume may be taken as 
representatives of the scientific wing. In Aristotle, 
Descartes, Berkeley, and Kant we find both groups of 
motives strongly present. 

Herbert Spencer, in whose honour we are assembled 
to-day, would naturally be classed among scientific 
philosophers : it was mainly from science that he drew 
his data, his formulation of problems, and his conception 
of method. But his strong religious sense is obvious 
in much, of his writing, and his ethical preoccupations 
are what make him value the conception of evolution 
that conception in which, as_a whole generation has 
believed, sd(m(x_andjnorajs^ are to be united in fruitful 
and indissoluble marriage. 
~~It is my belief That the ethical and religious motives 

97 



98 MYSTICISM AND LOGIC 

in spite of the splendidly imaginative systems to which 
they have given rise, have begn_on the whole a hindrance 
to the progress of philosophy, and ought now to be 
consciously thrust aside by those who wish to discover 
philosophical truth. Science, originally, was entangled 
in similar motives, and was thereby hindered in its 
advances. It is, I maintain, from science, rather than 
from ethics and religion, that philosophy should draw 
its inspiration. 

But there are two different ways in which a philosophy 
may seek to base itself upon science. It may emphasise 
the most general results^oi science, and see!Tto_giye~even 
greater generality and unity to these results. Or it may 
study the methods of science, and seek to apply these 
methods, with the necessary adaptations, to its own 
peculiar province. Much philosophy inspired by science 
has gone astray through preoccupation . _ with. jtha results 
momentarily supposed to have been achieved. It is not 
results, but (jnetho^ that can be transferred with profit 
from the sphere of the special sciences to the sphere of 
philosophy. What I wish to bring to your notice is the 
possibility and importance of applying to philosophical 
problems certain broad principles of method which have 
been found successful in the study of scientific questions. 

The opposition between a philosophy guided by 
scientific method and a philosophy dominated by religious 
and ethical ideas may be illustrated by two notions which 
are very prevalent in the works of philosophers, namely 
the notion _oi^thuniyrs&, and the notion oi good and 
eviL A philosopher is expected to tell us something about 
the nature of the universe as a whole, and to give grounds 
for either optimism or pessimism. Both these expecta 
tions seem to me mistaken. I believe the conception 
of " the universe " to be ; as its etymology indicates, a 



SCIENTIFIC METHOD IN PHILOSOPHY 99 

mere relic of pre-Copernican astronomy : and I believe 
the question of optimism and pessimism to be one which 
the philosopher will regard as outside his scope, except, 
possibly, to the extent of maintaining that it is insoluble. 

In the days before Copernicus, the conception of the 
" universe " was defensible on scientific grounds : the 
diurnal revolution of the heavenly bodies bound them 
together as all parts of one system, of which the earth 
was the centre. Round this apparent scientific fact, 
many human desires rallied : the wish to believe ManiL 
important in the scheme of things, the theoretical desirq q 
for a comprehensive understanding of the Whole, the 
hope that the course of nature might be guided by some 
sympathy with our wishes. In this way, an_ethically / 
inspired system of metaphysics grew up, whose anthro- 
pocentrism was apparently warranted by the geocentrism 
of astronomy. When Copernicus swept away the astrono 
mical basis of this system of thought, it had grown so 
familiar, and had associated itself so intimately with men s 
aspirations, that it survived with scarcely diminished 
force survived even Kant s " Copernican revolution," 
and is still now the unconscious premiss of most meta 
physical systems. 

The oneness of the world is an almost undiscussed 
postulate of most metaphysics. " Reality is not merely 
one and self -consistent, but is a system of reciprocally 
determinate parts" 1 such a statement would pass almost 
unnoticed as a mere truism. Yet I believe that it em 
bodies a failure to effect thoroughly the " Copernican 
revolution," and that the apparent oneness of the world 
is merely the oneness ofjwhat . js_seen by ji single spectator 
or apprehended by a single mind. The Critical Philosophy, 
although it intended to emphasise the subjective element 

1 Bosanquet, Lofio, il, p. 211. 



ioo MYSTICISM AND LOGIC 

in many apparent characteristics of the world, yet, by 
regarding the world in itself as unknowable, so con 
centrated attention upon the subjective representation 
that its subjectivity was soon forgotten. Having re 
cognised the categories as the work of the mind, it was 
paralysed by its own recognition, and abandoned in 
despair the attempt to undo the work of subjective 
falsification. In part, no doubt, its despair was well 
founded, but not, I think, in any absolute or ultimate 
sense. Still less was it a ground for rejoicing, or for 
supposing that the nescience to which it ought to have 
given rise could be legitimately exchanged for a meta 
physical dogmatism. 



As regards our present question, namely, the question 
of the unity of the world, the right method, as I think, 
has been indicated by William James. 1 " Let us now 
turn our backs upon ineffable or unintelligible ways 
of accounting for the world s oneness, and inquire whether, 
instead of being a principle, the oneness affirmed may 
not merely be a name like substance descriptive of 
the fact that certain specific and verifiable connections 
are found among the parts of the experiential flux. . . . 
We can easily conceive of things that shall have no connec 
tion whatever with each other. We may assume them 
to inhabit different times and spaces, as the dreams of 
different persons do even now. They may be so unlike 
and incommensurable, and so inert towards one another, 
as never to jostle or interfere. Even now there may 
actually be whole universes so disparate from ours that 
we who know ours have no means of perceiving that they 
exist. We conceive their diversity, however ; and by that 

1 Some ProlUms of Philosophy, p 124. 



SCIENTIFIC METHOD IN PHILOSOPHY 101 

fact the whole lot of them form what is known in logic 
as a universe of discourse/ To form a universe of 
discourse argues, as this example shows, no further kind 
of connexion. The importance attached by certain monistic 
writers to the fact that any chaos may become a universe 
by merely being named, is to me incomprehensible." 
We are thus left with two kinds of unity in the experienced 
world ; the one what we may call the epistemological 
unity, due merely to the fact that my experienced world 
is what one experience selects from the sum total of 
existence ; the other that tentative and partial unity 



exhibited in the prevalence of scientific laws in those 
portions of the world which science has hitherto mastered. 
Now a generalisation based upon either of these kinds of 
unity would be fallacious. That the things which we 
experience have the common property of being ex 
perienced by us is a truism from which obviously nothing 
of importance can be deducible : it is clearly fallacious 
to draw from the fact that whatever we experience is 
experienced the conclusion that therefore everything 
must be experienced. The generalisation of the second 
kind of unity, namely, that derived from scientific laws, 
would be equally fallacious, though the fallacy is a trifle 
less elementary. In order to explain it let us consider 
for a moment what is called thecfSgn oflaw^ People 
often speak as though it were a remarkable fact that the 
physical world is subject to invariable laws. In fact, 
however, it is not easy to see how such a world could & 
fail to obey general laws. Taking any arbitrary set .^ 
of points in space, there is a function of the time corre- \j. ^ 
sponding to these points, i.e. expressing the motion of a 
particle which traverses these points : this function may; 
be regarded as a general law to which the behaviour of 
such a particle is subject. Taking all such functions for 

H 



ioa MYSTICISM AND LOGIC 

all the particles in the universe, there will be theo 
retically some one formula embracing them all, and this 
formula may be regarded as the single and supreme law 
of the spatio-temporal world. Thus what is surprising 
in physics is not the existence of general laws, but their 
extreme simplicity. It is not the uniformity of nature 
that should surprise us, for, by sufficient analytic ingenuity, 
any conceivable course of nature might be shown to 
exhibit uniformity. What should sjarprise us is the 
fact that the uniformity is simple enough for us to be 
able to discover it. But it is just this characteristic 
of simplicity in the laws of nature hitherto discovered 
which it would be fallacious to generalise, for it is obvious 
that simplicity has been a part cause of their discovery, 
and can, therefore, give no ground for the supposition 
that other undiscovered laws are equally simple, 
ihe fallacies to which these two kinds of unity have 
given rise suggest a caution as regards all use in philoso 
phy of general results that science is supposed to have 
achieved. In the first place, in generalising these results 
beyond past experience, it is necessary to examine very 
carefully whether there is not some reason making it 
more probable that these results should hold of all that 
has been experienced than that they should hold of 
things universally J^Hie^sumtptal of what is experienced k 
by mankind is a selection from the sum total of what 
exists, and any general character exhibited by this 
selection may be due to the manner of selecting rather 
than to the general character of that from which ex- 
xT^perience selects. In the second place, the jnost genera 
results of science are the least certain and the most liable to 
be upset by subsequent research. In utilizing these results 
as the basis of a philosophy, we sacrifice the most valu 
able and remarkable characteristic of scientific method, 



SCIENTIFIC METHOD IN PHILOSOPHY 103 

namely, that, although almost every tiling in science is 
found sooner or later to require some correction, yet this 
correction is almost always such as to leave untouched, or 
only slightly modified, the greater part of the results 
which have been deduced from the premiss subsequently 
discovered to be faulty. The prudent man of science 
acquires a certain instinct as to the kind of uses which 
may be made of present scientific beliefs without incurring 
the danger of complete and utter refutation from the 
modifications likely to be introduced by subsequent 
discoveries. Unfortunately the use of scientific generalisa 
tions of a sweeping kind as the basis of philosophy is 
just that kind of use which an instinct of scientific caution 
would avoid, since, as a rule, it^oj:dd^qnly_lea4_to_true 
results if the generalisation upon which it is based stood 
in no need of correction. \ j JJT?U ff/V$ \A 

We may illustrate these general considerations by 
means of two examples, namely, the conservation of 
energy and the principle of evolution. 

(i) Let us begin with the conservation of energy, or, 
as Herbert Spencer used to call it, the persistence of 
force. He says i 1 

" Before taking a first step in the rational inter 
pretation of Evolution, it is needful to recognise, 
not only the facts that Matter is indestructible and 
Motion continuous, but also the fact that Force 
persists. An attempt to assign the causes of Evo 
lution would manifestly be absurd if that agency to 
which the metamorphosis in general and in detail 
is due, could either come into existence or cease to 
exist. The succession of phenomena would in such 
case be altogether arbitrary, and deductive Science 
impossible." 

1 First Principles (1862), Part II, beginning of chap. viu. 



104 MYSTICISM AND LOGIC 

This paragraph illustrates the kind of way in which 
t- ne philosopher is tempted to give an .air of absoluteness 
and necessity to empirical generalisations, of which only 
the approximate truth in the regions hitherto investi 
gated can be guaranteed by the unaided methods of 
science. It is very often said that the persistence of 
something or other is a necessary presupposition of all 
scientific investigation, and this presupposition is then 
thought to be exemplified in some quantity which 
physics declares to be constant. There are here, as it 
seems to me, three distinct errors. First, the detailed 
scientific investigation of nature does not presuppose any 
such general laws as its results are found to verify. 
Apart from particular observations, science need pre 
suppose nothing except the general principles of logic, 
and these principles are not laws of nature, for they are 
merely hypothetical, and apply not only to the actual 
world but to whatever is possible. The second error 
consists in the identification of a constant quantity with 
a persistent entity. Energy is a certain function of 
a physical system, but is not a thing or substance per 
sisting throughout the changes of the system. The same 
is true of mass, in spite of the fact that mass has often 
been defined as quantity of matter. The whole conception, 
of quantity, involving, as it does, numerical measurement 
based largely upon conventions, is far more artificial, 
far more an embodiment of mathematical convenience, 
than is commonly believed by those who philosophise 
on physics. Thus even if (which I cannot for a moment 
admit) the persistence of some entity were among the 
necessary postulates of science, it would be a sheer error 
to infer from this the constancy of any physical quantity, 
or the a priori necessity of any such constancy which 
may be empirically discovered. In the third place, it 



SCIENTIFIC METHOD IN PHILOSOPHY 105 

has become more and more evident with the progress of 
physics that large generalisations, such as the conserva 
tion of energy or mass, are far from certain and are 
very likely only approximate. Mass, which used to be 
regarded as the most indubitable of physical quantities, 
is now generally believed to vary according to velocity, 
and to be, in fact, a vector quantity which at a 
given moment is different in different directions. The 
detailed conclusions deduced from the supposed constancy 
of mass for such motions as used to be studied 
in physics will remain very nearly exact, and therefore 
over the field of the older investigations very little modi 
fication of the older results is required. But as soon as 
such a principle as the conservation of mass or of energy 
is erected into a universal a priori law, the slightest 
failure in absolute exactness is fatal, and the whole 
philosophic structure raised upon this foundation is 
necessarily ruined. The prudent philosopher, there 
fore, though he may with advantage study the 
methods of physics, will be very chary of basing 
anything upon what happen at the moment to be 
the most general results apparently obtained by those 
methods. 

(2) The philosophy of evolution, which was to be our 
second example, illustrates the same tendency to hasty 
generalisation, and also another sort, namely, the undue 
preoccupation with ethical notions. There are two 
kinds of evolutionist philosophy, of which both Hegel 
and Spencer represent the older and less radical kind, 
while Pragmatism and Bergson represent the more 
modern and revolutionary variety. But both these sorts 
of evolutionism have in common the emphasis on progress, 
that is, upon a continual change from the worse to the 
better, or from the simpler to the more complex. It 



106 MYSTICISM AND LOGIC 

would be unfair to attribute to Hegel any scientific 
motive or foundation, but all the other evolutionists, 
including Hegel s modern disciples, have derived their 
impetus very largely from the history of biological 
development. To a philosophy which derives a law of 
universal progress from this history there are two objec 
tions. First, that this history itself is concerned with a 
very small selection of facts confined to an infinitesimal 
fragment of space and time, and even on scientific 
grounds probably not an average sample of events 
in the world at large. For we know that decay 
as well as growth is a normal occurrence in the world. 
An extra-terrestrial philosopher, who had watched 
a single youth up to the age of twenty-one and had never 
come across any other human being, might conclude that 
it is the nature of human beings to grow continually 
taller and wiser in an indefinite progress towards per 
fection ; and this generalisation would be just as well 
founded as the generalisation which evolutionists base 
upon the previous history of this planet. Apart, how 
ever, from this scientific objection to evolutionism, 
there is another, derived from the undue admixture 
of ethical notions in the very idea of progress from which 
evolutionism derives its charm. Organic life, we are told, 
has developed gradually from the protozoon to the 
philosopher, and this development, we are assured, is 
indubitably an advance. Unfortunately it is the philoso 
pher, not the protozoon, who gives us this assurance, 
and we can have no security that the impartial outsider 
would agree with the philosopher s self-complacent 
assumption. This point has been illustrated by the 
philosopher Chuang Tzii in the following instructive 
anecdote : 



SCIENTIFIC METHOD IN PHILOSOPHY 107 

" The Grand Augur, in his ceremonial robes, ap 
proached the shambles and thus addressed the pigs : 
How can you object to die ? I shall fatten you for 
three months. I shall discipline myself for ten days 
and fast for three. I shall strew fine grass, and place 
you bodily upon a carved sacrificial dish. Does not 
this satisfy you ? 

Then, speaking from the pigs point of view, he 
continued : It is better, perhaps, after all, to live on 
bran and escape the shambles. . . . 

But then, added he, speaking from his own point 
of view, to enjoy honour when alive one would 
readily die on a war-shield or in the headsman s basket. 

So he rejected the pigs point of view and adopted 
his own point of view. In what sense, then, was he 
different from the pigs ? " 

I much fear that the evolutionists too often resemble 
the Grand Augur and the pigs. 

The ethical element which has been prominent in 
many of the most famous systems of philosophy is, in 
my opinion, one of the most serious obstacles to the 
victory of scientific method in the investigation of philo 
sophical questions. Human ethical notions, as Chuang 
Tzu perceived, are essentially anthropocentric, and 
involve, when used in metaphysics, an attempt, how 
ever veiled, to legislate for the universe on the basis of the 
present desires of men. In this way they interfere with 
that receptivity to fact which is the essence of the 
scientific attitude towards the world. To regard ethical 
notions as a key to the understanding of the world is 
essentially pre-Copernican. It is to make man, with the 
hopes and ideals which he happens to have at the present 
moment, the centre of the universe and the interpreter of 
its supposed aims and purposes. Ethical metaphysics 
is fundamentally an attempt, however disguised, to 



io8 MYSTICISM AND LOGIC 

give legislative force to our own wishes. This may, of 
course, be questioned, but I think that it is confirmed by 
a consideration of the way in which ethical notions arise. 
Ethics is essentially a product of the gregarious instinct, 
that is to say, of the instinct to co-operate with those 
who are to form our own group against those who belong 
to other groups. Those who belong to our own group 
are good ; those who belong to hostile groups are wicked. 
The ends which are pursued by our own group are desir 
able ends, the ends pursued by hostile groups are nefari 
ous. The subjectivity of this situation is not apparent 
to the gregarious animal, which feels that the general 
principles of justice are on the side of its own herd. 
When the animal has arrived at the dignity of the meta 
physician, it invents ethics as the embodiment of its 
belief in the justice of its own herd. So the Grand 
Augur invokes ethics as the justification of Augurs in 
their conflicts with pigs. But, it may be said, this view 
of ethics takes no account of such truly ethical notions as 
that of self-sacrifice. This, however, would be a mistake. 
The success of gregarious animals in the struggle for 
existence depends upon co-operation within the herd, and 
co-operation requires sacrifice, to some extent, of what 
would otherwise be the interest of the individual. Hence 
arises a conflict of desires and instincts, since both self- 
preservation and the preservation of the herd are biological 
ends to the individual. Ethics is in origin the art of 
recommending to others the sacrifices required for co-oper 
ation with oneself. Hence, by reflexion, it comes, through 
the operation of social justice, to recommend sacrifices 
by oneself, but all ethics, however refined, remains more 
or less subjective. Even vegetarians do not hesitate, 
for example, to save the life of a man in a fever, although 
in doing so they destroy the lives of many millions of 



SCIENTIFIC METHOD IN PHILOSOPHY 109 

microbes. The view of the world taken by the philosophy 
derived from ethical notions is thus never impartial 
and therefore never fully scientific. As compared with 
science, it fails to achieve the imaginative liberation from 
self which is necessary to such understanding of the 
world as man can hope to achieve, and the philosophy 
which it inspires is always more or less parochial, 
more or less infected with the prejudices of a time and 
a place. 

I do not deny the importance or value, within its own 
sphere, of the kind of philosophy which is inspired by 
ethical notions. The ethical work of Spinoza, for ex 
ample, appears to me of the very highest significance, 
but what is valuable in such work is not any meta 
physical theory as to the nature of the world to which 
it may give rise, nor indeed anything which can be 
proved or disproved by argument. What is valuable is 
the indication of some new way of feeling towards life 
and the world, some way of feeling by which our own 
existence can acquire more of the characteristics which 
we must deeply desire. The value of such work, how 
ever immeasurable it is, belongs with practice and not 
with theory. Such theoretic importance as it may 
possess is only in relation to human nature, not in re 
lation to the world at large. The scientific philosophy, 
therefore, which aims only at understanding the world 
and not directly at any other improvement of human 
life, cannot take account of ethical notions without being 
turned aside from that submission to fact which is the 
essence of the scientific temper. 



no MYSTICISM AND LOGIC 

II 

If the notion of the universe and the notion of good 
and evil are extruded from scientific philosophy, it may 
be asked what specific problems remain for the philos 
opher as opposed to the man of science ? It would be 
difficult to give a precise answer to this question, but 
certain characteristics may be noted as distinguishing 
the province of philosophy from that of the special 
sciences. 

In the first place a philosophical proposition must be 
general. It must not deal specially with things on the 
surface of the earth, or with the solar system, or with 
any other portion of space and time. It is this need of 
generality which has led to the belief that philosophy 
deals with the universe as a whole. I do not believe 
that this belief is justified, but I do believe that a philo 
sophical proposition must be applicable to everything 
that exists or may exist. It might be supposed that this 
admission would be scarcely distinguishable from the 
view which I wish to reject. This, however, would be 
an error, and an important one. The traditional view 
would make the universe itself the subject of various 
predicates which could not be applied to any particular 
thing in the universe, and the ascription of such peculiar 
predicates to the universe would be the special business 
of philosophy. I maintain, on the contrary, that there 
are no propositions of which the " universe " is the sub 
ject ; in other words, that there is no such thing as the 
" universe." What I do maintain is that there are 
general propositions which may be asserted of each 
individual thing, such as the propositions of logic. This 
does not involve that all the things there are form a whole 
which could be regarded as another thing and be made 



SCIENTIFIC METHOD IN PHILOSOPHY in 

the subject of predicates. It involves only the assertion 
that there are properties which belong to each separate 
thing, not that there are properties belonging to the 
whole of things collectively. The philosophy which 
I wish to advocate may be called logical atomism or 
absolute pluralism, because, while maintaining that 
there are many things, it denies that there is a whole 
composed of those things. We shall see, therefore, that 
philosophical propositions, instead of being concerned 
with the whole of things collectively, are concerned with 
all things distributively ; and not only must they be 
concerned with all things, but they must be concerned 
with such properties of all things as do not depend upon 
the accidental nature of the things that there happen to 
be, but are true of any possible world, independently of 
such facts as can only be discovered by our senses. 

This brings us to a second charateristic of philo 
sophical propositions, namely, that they must be a 
priori. A philosophical proposition must be such as can 
be neither proved nor disproved by empirical evidence. 
Too often we find in philosophical books arguments 
based upon the course of history, or the convolutions of 
the brain, or the eyes of shell-fish. Special and accidental 
facts of this kind are irrelevant to philosophy, which must 
make only such assertions as would be equally true 
however the actual world were constituted. 

We may sum up these two characteristics of philo 
sophical propositions by saying that philosophy is the 
science of the possible. But this statement unexplained 
is liable to be misleading, since it may be thought that 
the possible is something other than the general, whereas 
in fact the two are indistinguishable. 

Philosophy, if what has been said is correct, becomes 
indistinguishable from logic as that word has now come 



112 MYSTICISM AND LOGIC 

to be used. The study of logic consists, broadly speak 
ing, of two not very sharply distinguished portions. On 
the one hand it is concerned with those general state 
ments which can be made concerning everything without 
mentioning any one thing or predicate or relation, such 
for example as " if x is a member of the class a and every 
member of a is a member of ft , then x is a member of 
the class ft, whatever x, a, and ft may be." On the other 
hand, it is concerned with the analysis and enumeration 
of logical forms, i.e. with the kinds of propositions that 
may occur, with the various types of facts, and with the 
classification of the constituents of facts. In this way 
logic provides an inventory of possibilities, a repertory 
of abstractly tenable hypotheses. 

It might be thought that such a study would be too 
vague and too general to be of any very great importance, 
and that, if its problems became at any point sufficiently 
definite, they would be merged in the problems of some 
special science. It appears, however, that this is not the 
case. In some problems, for example, the analysis of 
space and time, the nature of perception, or the theory 
of judgment, the discovery of the logical form of the 
facts involved is the hardest part of the work and the 
part whose performance has been most lacking hitherto. 
It is chiefly for want of the right logical hypothesis that 
such problems have hitherto been treated in such an un 
satisfactory manner, and have given rise to those con 
tradictions or antinomies in which the enemies of reason 
among philosophers have at all times delighted. 

By concentrating attention upon the investigation of 
logical forms, it becomes possible at last for philosophy 
to deal with its problems piecemeal, and to obtain, as 
the sciences do, such partial and probably not wholly 
correct results as subsequent investigation can utilise 



SCIENTIFIC METHOD IN PHILOSOPHY 113 

even while it supplements and improves them. Most 
philosophies hitherto have been constructed all in one 
block, in such a way that, if they were not wholly correct, 
they were wholly incorrect, and could not be used as a 
basis for further investigations. It is chiefly owing to 
this fact that philosophy, unlike science, has hitherto been 
unprogressive, because each original philosopher has had 
to begin the work again from the beginning, without being 
able to accept anything definite from the work of his 
predecessors. A scientific philosophy such as I wish to 
recommend will be piecemeal and tentative like other 
sciences ; above all, it will be able to invent hypotheses 
which, even if they are not wholly true, will yet remain 
fruitful after the necessary corrections have been made. 
This possibility of successive approximations to the truth 
is, more than anything else, the source of the triumphs 
of science, and to transfer this possibility to philosophy 
is to ensure a progress in method whose importance 
it would be almost impossible to exaggerate. 

The essence of philosophy as thus conceived is analy 
sis, not synthesis. To build up systems of the world, like 
Heine s German professor who knit together fragments of 
life and made an intelligible system out of them, is not, 
I believe, any more feasible than the discovery of the 
philosopher s stone. What is feasible is the understanding 
of general forms, and the division of traditional problems 
into a number of separate and less baffling questions. 
" Divide and conquer " is the maxim of success here as 
elsewhere. 

Let us illustrate these somewhat general maxims by 
examining their application to the philosophy of space, 
for it is only in application that the meaning or impor 
tance of a method can be understood. Suppose we are 
confronted with the problem of space as presented in 



H4 MYSTICISM AND LOGIC 

Kant s Transcendental ^Esthetic, and suppose we wish 
to discover what are the elements of the problem and 
what hope there is of obtaining a solution of them. It 
will soon appear that three entirely distinct problems, 
belonging to different studies, and requiring different 
methods for their solution, have been confusedly combined 
in the supposed single problem with which Kant is 
concerned. There is a problem of logic, a problem of 
physics, and a problem of theory of knowledge. Of 
these three, the problem of logic can be solved exactly 
and perfectly ; the problem of physics can probably be 
solved with as great a degree of certainty and as great 
an approach to exactness as can be hoped in an empirical 
region ; the problem of theory of knowledge, however, 
remains very obscure and very difficult to deal with. 
Let us see how these three problems arise. 

(i) The logical problem has arisen through the 
suggestions of non-Euclidean geometry. Given a body 
of geometrical propositions, it is not difficult to find 
a minimum statement of the axioms from which this 
body of propositions can be deduced. It is also not 
difficult, by dropping or altering some of these axioms, 
to obtain a more general or a different geometry, having, 
from the point of view of pure mathematics, the same 
logical coherence and the same title to respect as the 
more familiar Euclidean geometry. The Euclidean 
geometry itself is true perhaps of actual space (though 
this is doubtful), but certainly of an infinite number of 
purely arithmetical systems, each of which, from the 
point of view of abstract logic, has an equal and inde 
feasible right to be called a Euclidean space. Thus 
space as an object of logical or mathematical study loses 
its uniqueness ; not only are there many kinds of spaces, 
but there are an infinity of examples of each kind, 



SCIENTIFIC METHOD IN PHILOSOPHY 115 

though it is difficult to find any kind of which the space 
of physics may be an example, and it is impossible to 
find any kind of which the space of physics is certainly 
an example. As an illustration of one possible logical 
system of geometry we may consider all relations of 
three terms which are analogous in certain formal respects 
to the relation " between " as it appears to be in actual 
space. A space is then defined by means of one such 
three-term relation. The points of the space are all the 
terms which have this relation to something or other, 
and their order in the space in question is determined 
by this relation. The points of one space are necessarily 
also points of other spaces, since there are necessarily 
other three-term relations having those same points for 
their field. The space in fact is not determined by the 
class of its points, but by the ordering three-term rela 
tion. When enough abstract logical properties of such 
relations have been enumerated to determine the resulting 
kind of geometry, say, for example, Euclidean geometry, 
it becomes unnecessary for the pure geometer in his ab 
stract capacity to distinguish between the various relations 
which have all these properties. He considers the whole 
class of such relations, not any single one among them. 
Thus in studying a given kind of geometry the pure 
mathematician is studying a certain class of relations 
defined by means of certain abstract logical properties 
which take the place of what used to be called axioms. 
The nature of geometrical reasoning therefore is purely"] 
deductive and purely logical ; if any special epistemolo- / 
gical peculiarities are to be found in geometry, it musti 
not be in the reasoning, but in our knowledge concerning} 
the axioms in some given space. 

(2) The physical problem of space is both more in 
teresting and more difficult than the logical problem. 



n6 MYSTICISM AND LOGIC 

The physical problem may be stated as follows : to find 
in the physical world, or to construct from physical 
materials, a space of one of the kinds enumerated by the 
logical treatment of geometry. This problem derives 
its difficulty from the attempt to accommodate to the 
roughness and vagueness of the real world some system 
possessing the logical clearness and exactitude of pure 
mathematics. That this can be done with a certain 
degree of approximation is fairly evident If I see three 
people A, B, and C sitting in a row, I become aware of 
the fact which may be expressed by saying that B is be 
tween A and C rather than that A is between B and C, 
or C is between A and B. This relation of " between " 
which is thus perceived to hold has some of the abstract 
logical properties of those three-term relations which, 
we saw, give rise to a geometry, but its properties fail to 
be exact, and are not, as empirically given, amenable 
to the kind of treatment at which geometry aims. In 
abstract geometry we deal with points, straight lines, and 
planes ; but the three people A, B, and C whom I see 
sitting in a row are not exactly points, nor is the row 
exactly a straight line. Nevertheless physics, which 
formally assumes a space containing points, straight 
lines, and planes, is found empirically to give results 
applicable to the sensible world. It must therefore be 
possible to find an interpretation of the points, straight 
lines, and planes of physics in terms of physical data, or 
at any rate in terms of data together with such hypo 
thetical additions as seem least open to question. Since 
all data suffer from a lack of mathematical precision 
through being of a certain size and somewhat vague in 
outline, it is plain that if such a notion as that of a point 
is to find any application to empirical material, the point 
must be neither a datum nor a hypothetical addition te 



SCIENTIFIC METHOD IN PHILOSOPHY 1.17 

da.ta, but a construction by means of data with their 
hypothetical additions. It is obvious that any hypo 
thetical filling out of data is less dubious and unsatis 
factory when the additions are closely analogous to data 
than when they are of a radically different sort. To 
assume, for example, that objects which we see continue, 
after we have turned away our eyes, to be more or less 
analogous to what they were while we were looking, is 
a less violent assumption than to assume that such objects 
are composed of an infinite number of mathematical 
points. Hence in the physical study of the geometry 
of physical space, points must not be assumed db initio as 
they are in the logical treatment of geometry, but must 
be constructed as systems composed of data and hypo 
thetical analogues of data. We are thus led naturally 
to define a physical point as a certain class of those 
objects which are the ultimate constituents of the physical 
world. It will be the class of all those objects which, as 
one would naturally say, contain the point. To secure a 
definition giving this result, without previously assuming 
that physical objects are composed of points, is an agree 
able problem in mathematical logic. The solution of 
this problem and the perception of its importance are 
due to my friend Dr. Whitehead. The oddity of regard 
ing a point as a class of physical entities wears off with 
familiarity, and ought in any case not to be felt by those 
who maintain, as practically every one does, that points 
are mathematical fictions. The word " fiction " is used 
glibly in such connexions by many men who seem not 
to feel the necessity of explaining how it can come about 
that a fiction can be so useful in the study of the actual 
world as the points of mathematical physics have been 
found to be. By our definition, which regards a point 
as a class of physical objects, it is explained both how 



u8 MYSTICISM AND LOGIC 

the use of points can lead to important physical results, 
and how we can nevertheless avoid the assumption that 
points are themselves entities in the physical world. 

Many of the mathematically convenient properties ol 
abstract logical spaces cannot be either known to belong 
or known not to belong to the space of physics. Such 
are all the properties connected with continuity. For 
to know that actual space has these properties would 
require an infinite exactness of sense-perception. If 
actual space is continuous, there are nevertheless many 
possible non-continuous spaces which will be empirically 
indistinguishable from it ; and, conversely, actual space 
may be non-continuous and yet empirically indistinguish 
able from a possible continuous space. Continuity, 
therefore, though obtainable in the a priori region ol 
arithmetic, is not with certainty obtainable in the space 
or time of the physical world : whether these are con 
tinuous or not would seem to be a question not only 
unanswered but for ever unanswerable. From the point 
of view of philosophy, however, the discovery that 
a question is unanswerable is as complete an answer as 
any that could possibly be obtained. And from the 
point of view of physics, where no empirical means of 
distinction can be found, there can be no empirical 
objection to the mathematically simplest assumption, 
which is that of continuity. 

The subject of the physical theory of space is a very 
large one, hitherto little explored. It is associated with 
a similar theory of time, and both have been forced upon 
the attention of philosophically minded physicists by the 
discussions which have raged concerning the theory of 
relativity. 

(3) The problem with which Kant is concerned in the 
Transcendental ^Esthetic is primarily the epistemological 



SCIENTIFIC METHOD IN PHILOSOPHY 119 

problem : " How do we come to have knowledge of 
geometry a priori ? " By the distinction between the 
logical and physical problems of geometry, the bearing 
and scope of this question are greatly altered. Our 
knowledge of pure geometry is a priori but is wholly 
logical. Our knowledge of physical geometry is synthetic, 
but is not a priori. Our knowledge of pure geometry 
is hypothetical, and does not enable us to assert, for 
example, that the axiom of parallels is true in the physical 
world. Our knowledge of physical geometry, while it 
does enable us to assert that this axiom is approximately 
verified, does not, owing to the inevitable inexactitude 
of observation, enable us to assert that it is verified 
exactly. Thus, with the separation which we have made 
between pure geometry and the geometry of physics, the 
Kantian problem collapses. To the question, " How 
is synthetic a priori knowledge possible ? " we can 
now reply, at any rate so far as geometry is concerned* 
"It is not possible, * if " synthetic " means " not de- 
ducible from logic alone." Our knowledge of geometry, 
like the rest of our knowledge, is derived partly from 
logic, partly from sense, and the peculiar position which 
in Kant s day geometry appeared to occupy is seen now 
to be a delusion. There are still some philosophers, it is 
true, who maintain that our knowledge that the axiom of 
parallels, for example, is true of actual space, is not to 
be accounted for empirically, but is as Kant maintained 
derived from an a priori intuition. This position is not 
logically refutable, but I think it loses all plausibility as 
soon as we realise how complicated and derivative is 
the notion of physical space. As we have seen, the 
application of geometry to the physical world in no way 
demands that there should really be points and straight 
lines among physical entities. The principle of economy, 



iao MYSTICISM AND LOGIC 

therefore, demands that we should abstain from assum 
ing the existence of points and straight lines. As soon, 
however, as we accept the view that points and straight 
lines are complicated constructions by means of classes 
of physical entities, the hypothesis that we have an 
a priori intuition enabling us to know what happens to 
straight lines when they are produced indefinitely becomes 
extremely strained and harsh ; nor do I think that such 
an hypothesis would ever have arisen in the mind of a 
philosopher who had grasped the nature of physical 
space. Kant, under the influence of Newton, adopted, 
though with some vacillation, the hypothesis of absolute 
space, and this hypothesis, though logically unobjection 
able, is removed by Occam s razor, since absolute space 
is an unnecessary entity in the explanation of the physical 
world. Although, therefore, we cannot refute the Kantian 
theory of an a priori intuition, we can remove its grounds 
one by one through an analysis of the problem. Thus, here 
as in many other philosophical questions, the analytic 
method, while not capable of arriving at a demonstrative 
result, is nevertheless capable of showing that all the 
positive grounds in favour of a certain theory are fallacious 
and that a less unnatural theory is capable of accounting 
for the facts. 

Another question by which the capacity of the analytic 
method can be shown is the question of realism. Both 
those who advocate and those who combat realism seem 
to me to be far from clear as to the nature of the problem 
which they are discussing. If we ask : " Are our objects 
of perception real and are they independent of the per 
cipient ? " it must be supposed that we attach some 
meaning to the words " real " and " independent," and 
yet, if either side in the controversy of realism is 
asked to define these two words, their answer is pretty 



SCIENTIFIC METHOD IN PHILOSOPHY 121 

sure to embody confusions such as logical analysis will 
reveal. 

Let us begin with the word " real." There certainly are 
objects of perception, and therefore, if the question 
whether these objects are real is to be a substantial 
question, there must be in the world two sorts of objects, 
namely, the real and the unreal, and yet the unreal is 
supposed to be essentially what there is not. The question 
what properties must belong to an object in order to 
make it real is one to which an adequate answer is seldom 
if ever forthcoming. There is of course the Hegelian 
answer, that the real is the self-consistent and that noth 
ing is self -consistent except the Whole ; but this answer, 
true or false, is not relevant in our present discussion, 
which moves on a lower plane and is concerned with the 
status of objects of perception among other objects of 
equal fragmentariness. Objects of perception are con 
trasted, in the discussions concerning realism, rather with 
psychical states on the one hand and matter on the other 
hand than with the all-inclusive whole of things. The 
question we have therefore to consider is the question 
as to what can be meant by assigning " reality " to some 
but not all of the entities that make up the world. Two 
elements, I think, make up what is felt rather than thought 
when the word " reality " is used in this sense. A thing 
is real if it persists at times when it is not perceived ; or 
again, a thing is real when it is correlated with other things 
in a way which experience has led us to expect. It will 
be seen that reality in either of these senses is by no 
means necessary to a thing, and that in fact there might 
be a whole world in which nothing was real in either of 
these senses. It might turn out that the objects of per 
ception failed of reality in one or both of these respects, 
without its being in any way deducible that they are 



122 MYSTICISM AND LOGIC 

not parts of the external world with which physics deals 
Similar remarks will apply to the word " independent." 
Most of the associations of this word are bound up with 
ideas as to causation which it is not now possible to 
maintain. A is independent of B when B is not an 
indispensable part of the cause of A. But when it is 
recognised that causation is nothing more than correla 
tion, and that there are correlations of simultaneity as 
well as of succession, it becomes evident that there is 
no uniqueness in a series of casual antecedents of a given 
event, but that, at any point where there is a correlation 
of simultaneity, we can pass from one line of antecedents 
to another in order to obtain a new series of causal 
antecedents. It will be necessary to specify the causal 
law according to which the antecedents are to be con 
sidered. I received a letter the other day from a corre 
spondent who had been puzzled by various philosophical 
questions. After enumerating them he says : " These 
questions led me from Bonn to Strassburg, where I found 
Professor Simmel." Now, it would be absurd to deny 
that these questions caused his body to move from 
Bonn to Strassburg, and yet it must be supposed that a 
set of purely mechanical antecedents could also be found 
which would account for this transfer of matter from one 
place to another. Owing to this plurality of causal series 
antecedent to a given event, the notion of the cause 
becomes indefinite, and the question of independence 
becomes correspondingly ambiguous. Thus, instead of 
asking simply whether A is independent of B, we ought 
to ask whether there is a series determined by such and 
such causal laws leading from B to A. This point is 
important in connexion with the particular question 
of objects of perception. It may be that no objects quite 
like those which we perceive ever exist unperceived ; 



SCIENTIFIC METHOD IN PHILOSOPHY 

in this case there will be a causal law according to which 
objects of perception are not independent of being 
perceived. But even if this be the case, it may never 
theless also happen that there are purely physical causal 
laws determining the occurrence of objects which are 
perceived by means of other objects which perhaps are 
not perceived. In that case, in regard to such causal 
laws objects of perception will be independent of being 
perceived. Thus the question whether objects of per 
ception are independent of being perceived is, as it 
stands, indeterminate, and the answer will be yes or no 
according to the method adopted of making it determinate. 
I believe that this confusion has borne a very large part 
in prolonging the controversies on this subject, which 
might well have seemed capable of remaining for ever 
undecided. The view which I should wish to advocate 
is that objects of perception do not persist unchanged 
at times when they are not perceived, although probably 
objects more or less resembling them do exist at such 
times ; that objects of perception are part, and the only 
empirically knowable part, of the actual subject-matter of 
physics, and are themselves properly to be called physical ; 
that purely physical laws exist determining the character 
and duration of objects of perception without any 
reference to the fact that they are perceived ; and that 
in the establishment of such laws the propositions of 
physics do not presuppose any propositions of psychology 
or even the existence of mind. I do not know whether 
realists would recognise such a view as realism. All 
that I should claim for it is, that it avoids difficulties 
which seem to me to beset both realism and idealism as 
hitherto advocated, and that it avoids the appeal which 
they have made to ideas which logical analysis shows 
to be ambiguous. A further defence and elaboration of 



124 MYSTICISM AND LOGIC 

the positions which I advocate, but for which time is 
lacking now, will be found indicated in my book on 
Our Knowledge of the External World* 

The adoption of scientific method in philosophy, if 
I am not mistaken, compels us to abandon the hope of 
solving many of the more ambitious and humanly 
interesting problems of traditional philosophy. Some 
of these it relegates, though with little expectation of 
a successful solution, to special sciences, others it shows 
to be such as our capacities are essentially incapable of 
solving. But there remain a large number of the re 
cognised problems of philosophy in regard to which the 
method advocated gives all those advantages of division 
into distinct questions, of tentative, partial, and pro 
gressive advance, and of appeal to principles with which, 
independently of temperament, all competent students 
must agree. The failure of philosophy hitherto has 
been due in the main to haste and ambition : patience 
and modesty, here as in other sciences, will open the 
road to solid and durable progress. 

1 Open Court Company, 1914. 



vii 

THE ULTIMATE CONSTITUENTS 
OF MATTER 1 

1WISH to discuss in this article no less a question 
than the ancient metaphysical query, " What is 
matter ? " The question, " What is matter ? " in so far 
as it concerns philosophy, is, I think, already capable of 
an answer which in principle will be as complete as an 
answer can hope to be ; that is to say, we can separate 
the problem into an essentially soluble and an essentially 
insoluble portion, and we can now see how to solve the 
essentially soluble portion, at least as regards its main 
outlines. It is these outlines which I wish to suggest in 
the present article. My main position, which is realistic, 
is, I hope and believe, not remote from that of Professor 
Alexander, by whose writings on this subject I have profited 
greatly. 2 It is also in close accord with that of Dr. Nunn. 8 
Common sense is accustomed to the division of the 
world into mind and matter. It is supposed by all who 
have never studied philosophy that the distinction be 
tween mind and matter is perfectly clear and easy, that 
the two do not at any point overlap, and that only a fool 
or a philosopher could be in doubt as to whether any 
given entity is mental or material. This simple faith 

1 An address delivered to the Philosophical Society of Manchester 
in February, 1915. Reprinted from The Monist, July, 1915. 

1 Cf. especially Samuel Alexander, " The Basis of Realism," British 
Academy, Vol. VI. 

* " Are Secondary Qualities Independent of Perception ? " Proc. 
Arist. Soc., 1909-10, pp. igi-2i8. 

1515 



MYSTICISM AND LOGIC 

survives in Descartes and in a somewhat modified form 
in Spinoza, but with Leibniz it begins to disappear, and 
from his day to our own almost every philosopher of note 
has criticised and rejected the dualism of common sense. 
It is my intention in this article to defend this dualism ; 
but before defending it we must spend a few moments on 
the reasons which have prompted its rejection. 

Our knowledge of the material world is obtained by 
means of the senses, of sight and touch and so on. At 
first it is supposed that things are just as they seem, but 
two opposite sophistications soon destroy this naive 
belief. On the one hand the physicists cut up matter 
into molecules, atoms, corpuscles, and as many more 
such subdivisions as their future needs may make them 
postulate, and the units at which they arrive are un 
commonly different from the visible, tangible objects of 
daily life. A unit of matter tends more and more to be 
something like an electromagnetic field filling all space, 
though having its greatest intensity in a small region. 
Matter consisting of such elements is as remote from 
daily life as any metaphysical theory. It differs from the 
theories of metaphysicians only in the fact that its 
practical efficacy proves that it contains some measure 
of truth and induces business men to invest money on the 
strength of it ; but, in spite of its connection with the money 
market, it remains a metaphysical theory none the less. 

The second kind of sophistication to which the world 
of common sense has been subjected is derived from the 
psychologists and physiologists. The physiologists point 
out that what we see depends upon the eye, that what we 
hear depends upon the ear, and that all our senses are 
liable to be affected by anything which affects the brain, 
like alcohol or hasheesh. Psychologists point out how 
much of what we think we see is supplied by association 



CONSTITUENTS OF MATTER 127 

or unconscious inference, how much is mental inter 
pretation, and how doubtful is the residuum which can 
be regarded as crude datum. From these facts it is 
argued by the psychologists that the notion of a datum 
passively received by the mind is a delusion, and it is 
argued by the physiologists that even if a pure datum of 
sense could be obtained by the analysis of experience, 
still this datum could not belong, as common sense sup 
poses, to the outer world, since its whole nature is con 
ditioned by our nerves and sense organs, changing as 
they change in ways which it is thought impossible to 
connect with any change in the matter supposed to be 
perceived. This physiologist s argument is exposed to 
the rejoinder, more specious than solid, that our know 
ledge of the existence of the sense organs and nerves is 
obtained by that very process which the physiologist has 
been engaged in discrediting, since the existence of the 
nerves and sense organs is only known through the 
evidence of the senses themselves. This argument may 
prove that some reinterpretation of the results of phy 
siology is necessary before they can acquire metaphysical 
validity. But it does not upset the physiological argu 
ment in so far as this constitutes merely a reductio ad 
absurdum of naive realism. 

These various lines of argument prove, I think, that 
some part of the beliefs of common sense must be aban 
doned. They prove that, if we take these beliefs as a 
whole, we are forced into conclusions which are in part 
self-contradictory ; but such arguments cannot of them 
selves decide what portion of our common-sense beliefs 
is in need of correction. Common sense believes that 
what we see is physical, outside the mind, and continuing 
to exist if we shut our eyes or turn them in another 
direction. I believe that common sense is right in 



128 MYSTICISM AND LOGIC 

regarding what we see as physical and (in one of 
several possible senses) outside the mind, but is 
probably wrong in supposing that it continues to exist 
when we are no longer looking at it. It seems to 
me that the whole discussion of matter has been obscured 
by two errors which support each other. The first of these 
is the error that what we see, or perceive through any of 
our other senses, is subjective : the second is the belief 
that what is physical must be persistent. Whatever 
physics may regard as the ultimate constituents of matter, 
it always supposes these constituents to be indestructible. 
Since the immediate data of sense are not indestructible 
but in a state of perpetual flux, it is argued that these 
data themselves cannot be among the ultimate con 
stituents of matter. I believe this to be a sheer mistake. 
The persistent particles of mathematical physics I regard 
as logical constructions, symbolic fictions enabling us to 
express compendiously very complicated assemblages of 
facts ; and, on the other hand, I believe that the actual 
data in sensation, the immediate objects of sight or touch 
or hearing, are extra-mental, purely physical, and among 
the ultimate constituents of matter. 

My meaning in regard to the impermanence of physical 
entities may perhaps be made clearer by the use of Berg- 
son s favourite illustration of the cinematograph. When 
I first read Bergson s statement that the mathematician 
conceives the world after the analogy of a cinematograph, 
I had never seen a cinematograph, and my first visit to 
one was determined by the desire to verify Bergson s 
statement, which I found to be completely true, at least 
so far as I am concerned. When, in a picture palace, we 
see a man rolling down hill, or running away from the 
police, or falling into a river, or doing any of those other 
things to which men in such places are addicted, we know 



CONSTITUENTS OF MATTER 129 

that there is riot really only one man moving, but a suc 
cession of films, each with a different momentary man. 
The illusion of persistence arises only through the ap 
proach to continuity in the series of momentary men. 
Now what I wish to suggest is that in this respect the 
cinema is a better metaphysician than common sense, 
physics, or philosophy. The real man too, I believe, 
however the police may swear to his identity, is really a 
series of momentary men, each different one from the 
other, and bound together, not by a numerical identity, 
but by continuity and certain intrinsic causal laws. And 
what applies to men applies equally to tables and chairs, 
the sun, moon and stars. Each of these is to be regarded, 
not as one single persistent entity, but as a series of 
entities succeeding each other in time, each lasting for a 
very brief period, though probably not for a mere mathe 
matical instant. In saying this I am only urging the 
same kind of division in time as we are accustomed to 
acknowledge in the case of space. A body which fills a 
cubic foot will be admitted to consist of many smaller 
bodies, each occupying only a very tiny volume ; similarly 
a thing which persists for an hour is to be regarded as 
composed of many things of less duration. A true theory 
of matter requires a division of things into time-corpuscles 
as well as into space-corpuscles. 

The world may be conceived as consisting of a multi 
tude of entities arranged in a certain pattern. The 
entities which are arranged I shall call " particulars." 
The arrangement or pattern results from relations among 
particulars. Classes or series of particulars, collected to 
gether on account of some property which makes it con 
venient to be able to speak of them as wholes, are what 
I call logical constructions or symbolic fictions. The par 
ticulars are to be conceived, not on the analogy of bricks 



i 3 o MYSTICISM AND LOGIC 

in a building, but rather on the analogy of notes in a 
sym phony. The ultimate constituents of a symphony 
(apart from relations) are the notes, each of which lasts 
only for a very short time. We may collect together 
all the notes played by one instrument : these may be 
regarded as the analogues of the successive particulars 
which common sense would regard as successive states of 
one " thing/ But the " thing " ought to be regarded as 
no more " real " or " substantial " than, for example, 
the role of the trombone. As soon as " things " are con 
ceived in this manner it will be found that the difficulties 
in the way of regarding immediate objects of sense as 
physical have largely disappeared. 

When people ask, " Is the object of sense mental or 
physical ? " they seldom have any clear idea either what 
is meant by " mental " or " physical/ or what criteria 
are to be applied for deciding whether a given entity 
belongs to one class or the other. I do not know how to 
give a sharp definition of the word " mental," but some 
thing may be done by enumerating occurrences which are 
indubitably mental : believing, doubting, wishing, willing, 
being pleased or pained, are certainly mental occurrences ; 
so are what we may call experiences, seeing, hearing, 
smelling, perceiving generally. But it does not follow 
from this that what is seen, what is heard, what is smelt, 
what is perceived, must be mental. When I see a flash 
of lightning, my seeing of it is mental, but what I see, 
although it is not quite the same as what anybody else 
sees at the same moment, and although it seems very 
unlike what the physicist would describe as a flash of 
lightning, is not mental. I maintain, in fact, that if the 
physicist could describe truly and fully all that occurs in 
the physical world when there is a flash of lightning, it 
would contain as a constituent what I see, and also what 



CONSTITUENTS OF MATTER 131 

is seen by anybody else who would commonly be said to 
see the same flash. What I mean may perhaps be made 
plainer by saying that if my body could remain in 
exactly the same state in which it is, although my mind 
had ceased to exist, precisely that object which I now see 
when I see the flash would exist, although of course I 
should not see it, since my seeing is mental. The prin 
cipal reasons which have led people to reject this view 
have, I think, been two : first, that they did not ade 
quately distinguish between my seeing and what I see ; 
secondly, that the causal dependence of what I see upon 
my body has made people suppose that what I see can 
not be " outside " me. The first of these reasons need 
not detain us, since the confusion only needs to be 
pointed out in order to be obviated ; but the second 
requires some discussion, since it can only be answered 
by removing current misconceptions, on the one hand as 
to the nature of space, and on the other, as to the mean 
ing of causal dependence. 

When people ask whether colours, for example, or 
other secondary qualities are inside or outside the mind, 
they seem to suppose that their meaning must be clear, 
and that it ought to be possible to say yes or no without 
any further discussion of the terms involved. In fact, 
however, such terms as " inside " or " outside " are very 
ambiguous. What is meant by asking whether this or 
that is " in " the mind ? The mind is not like a bag or a pie ; 
it does not occupy a certain region in space, or, if (in a sense) 
it does, what is in that region is presumably part of the 
brain, which would not be said to be in the mind. When 
people say that sensible qualities are in the mind, they 
do not mean " spatiaUy contained in " in the sense in 
which the blackbirds were in the pie. We might regard 
the mind as an assemblage of particulars, namely, what 



132 MYSTICISM AND LOGIC 

would be called " states of mind," which would belong 
together in virtue of some specific common quality. The 
common quality of all states of mind would be the quality 
designated by the word " mental " ; and besides this we 
should have to suppose that each separate person s 
states of mind have some common characteristic distin 
guishing them from the states of mind of other people. 
Ignoring this latter point, let us ask ourselves whether 
the quality designated by the word " mental " does, as a 
matter of observation, actually belong to objects of sense, 
such as colours or noises. I think any candid person 
must reply that, however difficult it may be to know what 
we mean by " mental/ it is not difficult to see that 
colours and noises are not mental in the sense of having 
that intrinsic peculiarity which belongs to beliefs and 
wishes and volitions, but not to the physical world. 
Berkeley advances on this subject a plausible argument 1 
which seems to me to rest upon an ambiguity in the word 
" pain." He argues that the realist supposes the heat 
which he feels in approaching a fire to be something 
outside his mind, but that as he approaches nearer and 
nearer to the fire the sensation of heat passes imper 
ceptibly into pain, and that no one could regard pain as 
something outside the mind. In reply to this argument, 
it should be observed in the first place that the heat of 
which we are immediately aware is not in the fire but in 
our own body. It is only by inference that the fire is 
judged to be the cause of the heat which we feel in our 
body. In the second place (and this is the more im 
portant point), when we speak of pain we may mean one 
of two things : we may mean the object of the sensation 
or other experience which has the quality of being painful, 

1 First dialogue between Hylas and Philonous, Works (Fraser s 
edition IQOT). I. p. 384. 



CONSTITUENTS OF MATTER 133 

or we may mean the quality of painfulness itself. When 
a man says he has a pain in his great toe, what he means 
is that he has a sensation associated with his great toe 
and having the quality of painfulness. The sensation 
itself, like every sensation, consists in experiencing a 
sensible object, and the experiencing has that quality of 
painfulness which only mental occurrences can have, but 
which may belong to thoughts or desires, as well as to 
sensations. But in common language we speak of the 
sensible object experienced in a painful sensation as a 
pain, and it is this way of speaking which causes the 
confusion upon which the plausibility of Berkeley s 
argument depends. It would be absurd to attribute the 
quality of painfulness to anything non-mental, and hence 
it comes to be thought that what we call a pain in the toe 
must be mental. In fact, however, it is not the sensible 
object in such a case which is painful, but the sensation, 
that is to say, the experience of the sensible object. As 
the heat which we experience from the fire grows greater, 
the experience passes gradually from being pleasant to 
being painful, but neither the pleasure nor the pain is a 
quality of the object experienced as opposed to the 
experience, and it is therefore a fallacy to argue that this 
object must be mental on the ground that painfulness can 
only be attributed to what is mental. 

If, then, when we say that something is in the mind 
we mean that it has a certain recognisable intrinsic 
characteristic such as belongs to thoughts and desires, it 
must be maintained on grounds of immediate inspection 
that objects of sense are not in any mind. 

A different meaning of "in the mind " is, however, to 
be inferred from the arguments advanced by those who 
regard sensible objects as being in the mind. The argu 
ments used are, in the main, such as would prove the 
J 



134 MYSTICISM AND LOGIC 

cpusal dependence of objects of sense upon the percipient. 
Now the notion of causal dependence is very obscure and 
difficult, much more so in fact than is generally realised 
by philosophers. I shall return to this point in a moment. 
For the present, however, accepting the notion of causal 
dependence without criticism, I wish to urge that the 
dependence in question is rather upon our bodies than 
upon our minds. The visual appearance of an object is 
altered if we shut one eye, or squint, or look previously 
at something dazzling ; but all these are bodily acts, and 
the alterations which they effect are to be explained by 
physiology and optics, not by psychology. 1 They are in 
fact of exactly the same kind as the alterations effected 
by spectacles or a microscope. They belong therefore to 
the theory of the physical world, and can have no bearing 
upon the question whether what we see is causally 
dependent upon the mind. What they do tend to prove, 
and what I for my part have no wish to deny, is that what 
we see is causally dependent upon our body arid is not, 
as crude common sense would suppose, something which 
would exist equally if our eyes and nerves and brain 
were absent, any more than the visual appearance pre 
sented by an object seen through a microscope would re 
main if the microscope were removed. So long as it is 
supposed that the physical world is composed of stable and 
more or less permanent constituents, the fact that what we 
see is changed by changes in our body appears to afford 
reason for regarding what we see as not an ultimate con 
stituent of matter. But if it is recognised that the ultimate 
constituents of matter are as circumscribed in duration as 
in spatial extent, the whole of this difficulty vanishes. 
There remains, however, another difficulty, connected 
space. When we look at the sun we wish to know 

This point has been well urged by the American realists 



CONSTITUENTS OF MATTER 135 

something about the sun itself, which is ninety-three 
million miles away ; but what we see is dependent upon 
our eyes, and it is difficult to suppose that our eyes can 
affect what happens at a distance of ninety-three million 
miles. Physics tells us that certain electromagnetic 
waves start from the sun, and reach our eyes after about 
eight minutes. They there produce disturbances in the 
rods and cones, thence in the optic nerve, thence in the 
brain. At the end of this purely physical series, by some 
odd miracle, comes the experience which we call " seeing 
the sun/ and it is such experiences which form the whole 
and sole reason for our belief in the optic nerve, the rods 
and cones, the ninety-three million miles, the electro 
magnetic waves, and the sun itself. It is this curious 
oppositeness of direction between the order of causation 
as affirmed by physics, and the order of evidence as 
revealed by theory of knowledge, that causes the most 
serious perplexities in regard to the nature of physical 
reality. Anything that invalidates our seeing, as a source 
of knowledge concerning physical reality, invalidates also 
the whole of physics and physiology. And yet, starting 
from a common-sense acceptance of our seeing, physics has 
been led step by step to the construction of the causal chain 
in which our seeing is the last link, and the immediate 
object which we see cannot be regarded as that initial cause 
which we believe to be ninety-three million miles away, and 
which we are inclined to regard as the " real " sun. 

I have stated this difficulty as forcibly as I can, be 
cause I believe that it can only be answered by a radical 
analysis and reconstruction of all the conceptions upon 
whose employment it depends. 

Space, time, matter and cause, are the chief of these 
conceptions. Let us begin with the conception of cause. 

Causal dependence, as I observed a moment ago, is a 



136 MYSTICISM AND LOGIC 

conception which it is very dangerous to accept at its face 
value. There exists a notion that in regard to any event 
there is something which may be called the cause of that 
event some one definite occurrence, without which the 
event would have been impossible and with which it be 
comes necessary. An event is supposed to be dependent 
upon its cause in some way which in it is not dependent 
upon other things. Thus men will urge that the mind is 
dependent upon the brain, or, with equal plausibility, that 
the brain is dependent upon the mind. It seems not im 
probable that if we had sufficient knowledge we could 
infer the state of a man s mind from the state of his brain, 
or the state of his brain from the state of his mind. So 
long as the usual conception of causal dependence is re 
tained, this state of affairs can be used by the materialist 
to urge that the state of our brain causes our thoughts, 
and by the idealist to urge that our thoughts cause the 
state of our brain. Either contention is equally valid or 
equally invalid. The fact seems to be that there are many 
correlations of the sort which may be called causal, and 
that, for example, either a physical or a mental event can 
be predicted, theoretically, either from a sufficient number 
of physical antecedents or from a sufficient number oi 
mental antecedents. To speak of the cause of an event is 
therefore misleading. Any set of antecedents from which 
the event can theoretically be inferred by means of correla 
tions might be called a cause of the event. But to speak oi 
the cause is to imply a uniqueness which does not exist. 

The relevance of this to the experience which we call 
" seeing the sun " is obvious. The fact that there exists 
a chain of antecedents which makes our seeing dependent 
upon the eyes and nerves and brain does not even tend to 
show that there is not another chain of antecedents in 
which the eyes and nerves and brain as physical things 
are ignored. If we are to escape from the dilemma which 



CONSTITUENTS OF MATTER 137 

seemed to arise out of the physiological causation of what 
we see when we say we see the sun, we must find, at least 
in theory, a way of stating causal laws for the physical 
world, in which the units are not material things, such as 
the eyes and nerves and brain, but momentary particulars 
of the same sort as our momentary visual object when we 
look at the sun. The sun itself and the eyes and nerves 
and brain must be regarded as assemblages of momentary 
particulars. Instead of supposing, as we naturally do 
when we start from an uncritical acceptance of the 
apparent dicta of physics, that matter is what is " really 
real " in the physical world, and that the immediate 
objects of sense are mere phantasms, we must regard 
matter as a logical construction, of which the con 
stituents will be just such evanescent particulars as 
may, when an observer happens to be present, become 
data of sense to that observer. What physics regards as 
the sun of eight minutes ago will be a whole assemblage 
of particulars, existing at different times, spreading out 
from a centre with the velocity of light, and containing 
among their number all those visual data which are seen 
by people who are now looking at the sun. Thus the sun 
of eight minutes ago is a class of particulars, and what I 
see when I now look at the sun is one member of this 
class. The various particulars constituting this class 
will be correlated with each other by a certain continuity 
and certain intrinsic laws of variation as we pass out 
wards from the centre, together with certain modifica 
tions correlated extrinsically with other particulars which 
are not members of this class. It is these extrinsic 
modifications which represent the sort of facts that, in 
our former account, appeared as the influence of the eyes 
and nerves in modifying the appearance of the sun. 1 

1 Cf. T. P. Nunn, " Are Secondary Qualities Independent of Per 
ception ? " Proc. Arist. Soc., 1900-1910. 



138 MYSTICISM AND LOGIC 

The prima facie difficulties in the way of this view are 
chiefly derived from an unduly conventional theory of 
space. It might seem at first sight as if we had packed the 
world much fuller than it could possibly hold. At every 
place between us and the sun, we said, there is to be a 
particular which is to be a member of the sun as it was a 
few minutes ago. There will also, of course, have to be a 
particular which is a member of any planet or fixed star 
that may happen to be visible from that place. At the 
place where I am, there will be particulars which will be 
members severally of all the " things " I am now said to 
be perceiving. Thus throughout the world, everywhere, 
there will be an enormous number of particulars co 
existing in the same place. But these troubles result 
from contenting ourselves too readily with the merely 
three-dimensional space to which schoolmasters have 
accustomed us. The space of the real world is a space of 
six dimensions, and as soon as we realise this we see that 
there is plenty of room for all the particulars for which 
we want to find positions. In order to realise this we 
have only to return for a moment from the polished space 
of physics to the rough and untidy space of our immediate 
sensible experience. The space of one man s sensible 
objects is a three-dimensional space. It does not appear 
probable that two men ever both perceive at the same 
time any one sensible object ; when they are said to see 
the same thing or hear the same noise, there will always 
be some difference, however slight, between the actual 
shapes seen or the actual sounds heard. If this is so, and 
if, as is generally assumed, position in space is purely 
relative, it follows that the space of one man s objects 
and the space of another man s objects have no place in 
common, that they are in fact different spaces, and not 
merely different parts of one space. I mean by this that 
such immediate spatial relations as are perceived to hold 



CONSTITUENTS OF MATTER 139 

between the different parts of the sensible space perceived 
by one man, do not hold between parts of sensible spaces 
perceived by different men. There are therefore a multi 
tude of three-dimensional spaces in the world : there are 
all those perceived by observers, and presumably also 
those which are not perceived, merely because no observer 
is suitably situated for perceiving them. 

But although these spaces do not have to one another 
the same kind of spatial relations as obtain between the 
parts of one of them, it is nevertheless possible to arrange 
these spaces themselves in a three-dimensional order 
This is done by means of the correlated particulars which 
we regard as members (or aspects) of one physical thing. 
When a number of people are said to see the same object, 
those who would be said to be near to the object see a 
particular occupying a larger part of their field of vision 
than is occupied by the corresponding particular seen by 
people who would be said to be farther from the thing. 
By means of such considerations it is possible, in ways 
which need not now be further specified, to arrange all 
the different spaces in a three-dimensional series. Since 
each of the spaces is itself three-dimensional, the whole 
world of particulars is thus arranged in a six-dimensional 
space, that is to say, six co-ordinates will be required to 
assign completely the position of any given particular, 
namely, three to assign its position in its own space and 
three more to assign the position of its space among the 
other spaces. 

There are two ways of classifying particulars : we may 
take together all those that belong to a given " perspec 
tive/ or all those that are, as common sense would say, 
different " aspects " of the same " thing." For example, 
if I am (as is said) seeing the sun, what I see belongs to 
two assemblages : (i) the assemblage of all my present 
objects of sense, which is *vhat 1 caiJ a " perspective " ; 



i 4 o MYSTICISM AND LOGIC 

(2) the assemblage of all the different particulars which 
would be called aspects of the sun of eight minutes 
ago this assemblage is what I define as being the sun oi 
eight minutes ago. Thus " perspectives " and " things " 
are merely two different ways of classifying particulars. It 
is to be observed that there is no a priori necessity for 
particulars to be susceptible of this double classification. 
There may be what might be called " wild " particulars, 
not having the usual relations by which the classification 
is effected ; perhaps dreams and hallucinations are 
composed of particulars which are " wild " in this sense. 

The exact definition of what is meant by a perspective 
is not quite easy. So long as we confine ourselves to 
visible objects or to objects of touch we might define the 
perspective of a given particular as " all particulars which 
have a simple (direct) spatial relation to the given par 
ticular." Between two patches of colour which I see 
now, there is a direct spatial relation which I equally see. 
But between patches of colour seen by different men 
there is only an indirect constructed spatial relation by 
means of the placing of " things " in physical space 
(which is the same as the space composed of perspec 
tives). Those particulars which have direct spatial 
relations to a given particular will belong to the same 
perspective. But if, for example, the sounds which I 
hear are to belong to the same perspective with the 
patches of colour which I see, there must be particulars 
which have no direct spatial relation and yet belong to 
the same perspective. We cannot define a perspective 
as all the data of one percipient at one time, because we 
wish to allow the possibility of perspectives which are 
not perceived by any one. There will be need, therefore, 
in defining a perspective, of some principle derived 
neither from psychology nor from space. 

Such a principle may be obtained from the considera- 



CONSTITUENTS OF MATTER 141 

tion of time. The one all-embracing time, like the one 
all-embracing space, is a construction ; there is no direct 
time-relation between particulars belonging to my per 
spective and particulars belonging to another man s. On 
the other hand, any two particulars of which I am aware 
are either simultaneous or successive, and their simul 
taneity or successiveness is sometimes itself a datum to 
me. We may therefore define the perspective to which a 
given particular belongs as " all particulars simultaneous 
with the given particular," where " simultaneous " is to 
be understood as a direct simple relation, not the deriva 
tive constructed relation of physics. It may be observed 
that the introduction of " local time " suggested by the 
principle of relativity has effected, for purely scientific 
reasons, much the same multiplication of times as we 
have just been advocating. 

The sum-total of all the particulars that are (directly) 
either simultaneous with or before or after a given par 
ticular may be defined as the " biography " to which that 
particular belongs. It will be observed that, just as a 
perspective need not be actually perceived by any one, 
so a biography need not be actually lived by any one. 
Those biographies that are lived by no one are called 
" official." 

The definition of a " thing " is effected by means of 
continuity and of correlations which have a certain 
differential independence of other " things." That is to 
say, given a particular in one perspective, there will 
usually in a neighbouring perspective be a very similar 
particular, differing from the given particular, to the first 
order of small quantities, according to a law involving 
only the difference of position of the two perspectives in 
perspective space, and not any of the other " things " in 
the universe. ( It is this continuity and differential in 
dependence in the law ef change as we pass from one 



i 4 2 MYSTICISM AND LOGIC 

perspective to another that defines the class oi particulars 
which is to be called " one thing." 

Broadly speaking, we may say that the physicist finds 
it convenient to classify particulars into " things," while 
the psychologist finds it convenient to classify them into 
" perspectives " and " biographies," since one perspective 
may constitute the momentary data of one percipient, and 
one biography may constitute the whole of the data of 
one percipient throughout his life. 

We may now sum up our discussion. Our object has 
been to discover as far as possible the nature of the 
ultimate constituents of the physical world. When I 
speak of the " physical world," I mean, to begin with, 
the world dealt with by physics. It is obvious that 
physics is an empirical science, giving us a certain amount 
of knowledge and based upon evidence obtained through 
the senses. But partly through the development of 
physics itself, party through arguments derived from 
physiology, psychology or metaphysics, it has come to 
be thought that the immediate data of sense could not 
themselves form part of the ultimate constituents of the 
physical world, but were in some sense " mental," " in 
the mind," or " subjective." The grounds for this view, 
in so far as they depend upon physics, can only be ade 
quately dealt with by rather elaborate constructions 
depending upon symbolic logic, showing that out of such 
materials as are provided by the senses it is possible to 
construct classes and series having the properties which 
physics assigns to matter. Since this argument is diffi 
cult and technical, I have not embarked upon it in this 
article. But in so far as the view that sense-data are 
" mental " rests upon physiology, psychology, or meta 
physics, I have tried to show that it rests upon con 
fusions and prejudices prejudices in favour of per 
manence in the ultimate constituents of matter, and 



CONSTITUENTS OF MATTER 143 

confusions derived from unduly simple notions as to 
space, from the causal correlation of sense-data with 
sense-organs, and from failure to distinguish between 
sense-data and sensations. If what we have said on 
these subjects is valid, the existence of sense-data is 
logically independent of the existence of mind, and is 
causally dependent upon the body of the percipient rather 
than upon his mind. The causal dependence upon the 
body of the percipient, we found, is a more complicated 
matter than it appears to be, and, like all causal depend 
ence, is apt to give rise to erroneous beliefs through mis 
conceptions as to the nature of causal correlation. If we 
have been right in our contentions, sense-data are merely 
those among the ultimate constituents of the physical 
world, of which we happen to be immediately aware ; 
they themselves are purely physical, and all that is mental 
in connection with them is our awareness of them, which 
is irrelevant to their nature and to their place in physics. 
Unduly simple notions as to space have been a great 
stumbling-block to realists. When two men look at the 
same table, it is supposed that what the one sees and 
what the other sees are in the same place. Since the 
shape and colour are not quite the same for the two men, 
this raises a difficulty, hastily solved, or rather covered 
up, by declaring what each sees to be purely " sub 
jective " though it would puzzle those who use this glib 
word to say what they mean by it. The truth seems to 
be that space and time also is much more complicated 
than it would appear to be from the finished structure of 
physics, and that the one all-embracing three-dimensional 
space is a logical construction, obtained by means of 
correlations from a crude space of six dimensions. The 
particulars occupying this six-dimensional space, classi 
fied in one way, form " things," from which with certain 
further manipulations we can obtain what physics can 



144 MYSTICISM AND LOGIC 

regard as matter ; classified in another way, they form 
" perspectives " and " biographies/ which may, if a 
suitable percipient happens to exist, form respectively 
the sense-data of a momentary or of a total experience. 
It is only when physical " things " have been dissected 
into series of classes of particulars, as we have done, that 
the conflict between the point of view of physics and the 
point of view of psychology can be overcome. This con 
flict, if what has been said is not mistaken, flows from 
different methods of classification, and vanishes as soon 
as its source is discovered. 

In favour of the theory which I have briefly outlined, 
I do not claim that it is certainly true. Apart from the 
likelihood of mistakes, much of it is avowedly hypo 
thetical. What I do claim for the theory is that it may 
be true, and that this is more than can be said for any 
other theory except the closely analogous theory of 
Leibniz. The difficulties besetting realism, the con 
fusions obstructing any philosophical account of physics, 
the dilemma resulting from discrediting sense-data, 
which yet remain the sole source of our knowledge of the 
outer world all these are avoided by the theory which I 
advocate. This does not prove the theory to be true, 
since probably many other theories might be invented 
which would have the same merits. But it does prove 
that the theory has a better chance of being true than 
any of its present competitors, and it suggests that what 
can be known with certainty is likely to be discoverable 
by taking our theory as a starting-point, and gradually 
freeing it from all such assumptions as seem irrelevant, 
unnecessary, or unfounded. On these grounds, I recom 
mend it to attention as a hypothesis and a basis for further 
work, though not as itself a finished or adequate solution 
of the problem with which it deals. 



vm 

THE RELATION OF SENSE-DATA 
TO PHYSICS 

I. THE PROBLEM STATED 

OHYSICS is said to be an empirical science, based 
A upon observation and experiment. 

It is supposed to be verifiable, i.e. capable of calcu 
lating beforehand results subsequently confirmed by 
observation and experiment. 

What can we learn by observation and experiment ? 

Nothing, so far as physics is concerned, except imme 
diate data of sense : certain patches of colour, sounds, 
tastes, smells, etc., with certain spatio-temporal rela 
tions. 

The supposed contents of the physical world are prima 
facie very different from these : molecules have no colour, 
atoms make no noise, electrons have no taste, and cor 
puscles do not even smell. 

If such objects are to be verified, it must be solely 
through their relation to sense-data : they must have 
some kind of correlation with sense-data, and must be 
verifiable through their correlation alone. 

But how is the correlation itself ascertained ? A cor 
relation can only be ascertained empirically by the cor 
related objects being constantly found together. But in 
our case, only one term of the correlation, namely, the 
sensible term, is ever found : the other term seems essen- 



i 4 6 MYSTICISM AND LOGIC 

tially incapable of being found. Therefore, it would seem, 
the correlation with objects of sense, by which physics was 
to be verified, is itself utterly and for ever un verifiable. 
There are two ways of avoiding this result. 

(1) We may say that we know some principle a -priori, 
without the need of empirical verification, e.g. that our 
sense-data have causes other than themselves, and that 
something can be known about these causes by inference 
from their effects. This way has been often adopted by 
philosophers. It may be necessary to adopt this way to 
some extent, but in so far as it is adopted physics ceases 
to be empirical or based upon experiment and observa 
tion alone. Therefore this way is to be avoided as much 
as possible. 

(2) We may succeed in actually defining the objects of 
physics as functions of sense-data. Just in so far as 
physics leads to expectations, this must be possible, since 
we can only expect what can be experienced. And in so 
far as the physical state of affairs is inferred from sense- 
data, it must be capable of expression as a function of 
sense-data. The problem of accomplishing this expres 
sion leads to much interesting logico-mathematical work. 

In physics as commonly set forth, sense-data appear 
as functions of physical objects : when such-and-such 
waves impinge upon the eye, we see such-and-such 
colours, and so on. But the waves are in fact inferred 
from the colours, not vice versa. Physics cannot be 
regarded as validly based upon empirical data until the 
waves have been expressed as functions of the colours 
and other sense-data. 

Thus if physics is to be verifiable we are faced with the 
following problem : Physics exhibits sense-data as func 
tions of physical objects, but verification is only possible 
if physical objects can be exhibited as functions of sense- 



SENSE-DATA AND PHYSICS 147 

data. We have therefore to solve the equations giving 
sense-data in terms of physical objects, so as to make 
them instead give physical objects in terms of sense- 
data. 

II. CHARACTERISTICS OF SENSE-DATA 

When I speak of a " sense-datum," I do not mean the 
whole of what is given in sense at one time. I mean 
rather such a part of the whole as might be singled out 
by attention : particular patches of colour, particular 
noises, and so on. There is some difficulty in deciding 
what is to be considered one sense-datum : often atten 
tion causes divisions to appear where, so far as can be 
discovered, there were no divisions before. An observed 
complex fact, such as that this patch of red is to the left 
of that patch of blue, is also to be regarded as a datum 
from our present point of view : epistemologically, it 
does not differ greatly from a simple sense-datum as 
regards its function in giving knowledge. Its logical 
structure is very different, however, from that of sense : 
sense gives acquaintance with particulars, and is thus a 
two-term relation in which the object can be named but 
not asserted, and is inherently incapable of truth or false 
hood, whereas the observation of a complex fact, which 
may be suitably called perception, is not a two-term 
relation, but involves the prepositional form on the 
object-side, and gives knowledge of a truth, not mere 
acquaintance with a particular. This logical difference, 
important as it is, is not very relevant to our present 
problem ; and it will be convenient to regard data of 
perception as included among sense-data for the purposes 
of this paper. It is to be observed that the particulars 
which are constituents of a datum of perception are 
always sense-data in the strict sense. 



148 MYSTICISM AND LOGIC 

Concerning sense-data, we know that they are there 
while they are data, and this is the epistemological basis 
of all our knowledge of external particulars. (The mean 
ing of the word " external " of course raises problems 
which will concern us later.) We do not know, except by 
means of more or less precarious inferences, whether the 
objects which are at one time sense-data continue to 
exist at times when they are not data. Sense-data at the 
times when they are data are all that we directly and 
primitively know of the external world ; hence in episte- 
mology the fact that they are data is all-important. But 
the fact that they are all that we directly know gives, of 
course, no presumption that they are all that there is. If 
we could construct an impersonal metaphysic, independent 
of the accidents of our knowledge and ignorance, the 
privileged position of the actual data would probably 
disappear, and they would probably appear as a rather 
haphazard selection from a mass of objects more or less 
like them. In saying this, I assume only that it is 
probable that there are particulars with which we are 
not acquainted. Thus the special importance of sense- 
data is in relation to epistemology, not to metaphysics. 
In this respect, physics is to be reckoned as metaphysics : 
it is impersonal, and nominally pays no special attention 
to sense-data. It is only when we ask how physics can 
be known that the importance of sense-data re-emerges. 



III. SENSIBILIA 

I shall give the name sensibilia to those objects which 
have the same metaphysical and physical status as sense- 
data, without necessarily being data to any mind. Thus 
the relation of a sensibile to a sense-datum is like that of 
a man to a husband : a man becomes a husband by 



SENSE-DATA AND PHYSICS 149 

entering into the relation of marriage, and similarly a 
sensibile becomes a sense-datum by entering into the 
relation of acquaintance. It is important to have both 
terms ; for we wish to discuss whether an object which 
is at one time a sense-datum can still exist at a time 
when it is not a sense-datum. We cannot ask " Can 
sense-data exist without being given ? " for that is like 
asking " Can husbands exist without being married ? " 
We must ask " Can sensibilia exist without being given ? " 
and also "Can a particular sensibile be at one time a 
sense-datum, and at another not ? " Unless we have the 
word sensibile as well as the word "sense-datum," such 
questions are apt to entangle us in trivial logical puzzles. 

It will be seen that all sense-data are sensibilia. It is 
a metaphysical question whether all sensibilia are sense- 
data, and an epistemological question whether there 
exist means of inferring sensibilia which are not data 
from those that are. 

A few preliminary remarks, to be amplified as we pro 
ceed, will serve to elucidate the use which I propose to 
make of sensibilia. 

I regard sense-data as not mental, and as being, in 
fact, part of the actual subject-matter of physics. There 
are arguments, shortly to be examined, for their sub 
jectivity, but these arguments seem to me only to prove 
physiologicaj_s\^]e.ciivity, i.e. causal dependence on the 
sense-organs, nerves, and brain. The appearance which 
a thing presents to us is causally dependent upon these, 
in exactly the same way as it is dependent upon inter 
vening fog or smoke or coloured glass. Both dependences 
are contained in the statement that the appearance 
which a piece of matter presents when viewed from a 
given place is a function not only of the piece of matter, 
but also of the. intervening medium. (The terms used in 



150 MYSTICISM AND LOGIC 

this statement " matter," " view from a given place," 
" appearance/ " intervening medium " will all be de 
nned in the course of the present paper.) We have not 
the means of ascertaining how things appear from places 
not surrounded by brain and nerves and sense-organs, 
br cause we cannot leave the body ; but continuity 
makes it not unreasonable to suppose that they present 
some appearance at such places. Any such appearance 
would be included among sensibilia. If $er impossibile 
there were a complete human body with no mind in 
side it, all those sensibilia would exist, in relation to that 
body, which would be sense-data if there were a mind in 
the body. What the mind adds to sensibilia, in fact, is 
merely awareness : everything else is physical or physio 
logical. 

IV. SENSE-DATA ARE PHYSICAL 

Before discussing this question it will be well to define 
the sense in which the terms " mental " and " physical " 
are to be used. The word " physical," in all preliminary 
discussions, is to be understood as meaning " what is 
dealt with by physics." Physics, it is plain, tells us some 
thing about some of the constituents of the actual world ; 
what these constituents are may be doubtful, but it is 
they that are to be called physical, whatever their nature 
may prove to be. 

The definition of the term " mental " is more difficult, 
and can only be satisfactorily given after many difficult 
controversies have been discussed and decided. For 
present purposes therefore I must content myself with 
assuming a dogmatic answer to these controversies. 1 
shall call a particular " mental " when it is aware of 
something, and I shall call a fact " mental " when it 
contains a mental particular as a constituent. 



SENSE-DATA AND PHYSICS 151 

It will be seen that the mental and the physical are not 
necessarily mutually exclusive, although I know of no 
reason to suppose that they overlap. 

The doubt as to the correctness of our definition of the 
" mental " is of little importance in our present dis 
cussion. For what I am concerned to maintain is that 
sense-data are physical, and this being granted it is a 
matter of indifference in our present inquiry whether or 
not they are also mental. Although I do npjt hold, with 
Mach and James and the "new realists," that the 
difference between the mental and the physical is merely 
one of arrangement, yet what I have to say in the present 
paper is compatible with their doctrine and might have 
been reached from their standpoint. 

In discussions on sense-data, two questions are com 
monly confused, namely : 

(i) Do sensible objects persist when we are not sensible 
of them ? in other words, do sensibilia which are data at a 
certain time some times continue to exist at times when they 
are not data ? And (2) are sense-data mental or physical ? 

I propose to assert that sense-data are physical, while 
yet maintaining that they probably never persist un 
changed after ceasing to be data. The view that they do 
not persist is often thought, quite erroneously in my 
opinion, to imply that they are mental ; and this has, 1 
believe, been a potent source of confusion in regard to 
our present problem. If there were, as some have held, 
a logical impossibility in sense-data persisting after ceasing 
to be data, that certainly would tend to show that they 
were mental ; but if, as I contend, their non-persistence 
is merely a probable inference from empirically ascer 
tained causal laws, then it carries no such implication 
with it, and we are quite free to treat them as part of the 
subject-matter of physics. 



152 MYSTICISM AND LOGIC 

Logically a sense-datum is an object, a particular of 
which the subject is aware. It does not contain the 
subject as a part, as for example beliefs and volitions do. 
The existence of the sense-datum is therefore not logically 
dependent upon that of the subject ; for the only way, 
so far as I know, in which the existence of A can be 
logically dependent upon the existence of B is when B 
is part of A. There is therefore no a priori reason why a 
particular which is a sense-datum should not persist 
after it has ceased to be a datum, nor why other similar 
particulars should not exist without ever being data. 
The view that sense-data are mental is derived, no doubt, 
in part from their physiological subjectivity, but in part 
also from a failure to distinguish between sense-data and 
" sensations." By a sensation I mean the fact consisting 
in the subject s awareness of the sense-datum. Thus a 
sensation is a complex of which the subject is a con 
stituent and which therefore is mental. The sense-datum, 
on the other hand, stands over against the subject as that 
external object of which in sensation the subject is 
aware. It is true that the sense-datum is in many cases 
in the subject s body, but the subject s body is as dis 
tinct from the subject as tables and chairs are, and is in 
fact merely a part of the material world. So soon, there 
fore, as sense-data are clearly distinguished from sensa 
tions, and as their subjectivity is recognised to be physio 
logical not psychical, the chief obstacles in the way of 
regarding them as physical are removed. 



V. " SENSIBILIA " AND 

But if " sensibilia " are to be recognised as the ultimate 
constituents of the physical world, a long and difficult 
journey is to be performed before we can arrive either at 



SENSE-DATA AND PHYSICS 153 

the " thing " of common sense or at the " matter " of 
physics. The supposed impossibility of combining the 
different sense-data which are regarded as appearances of 
the same " thing " to different people has made it seem 
as though these " sensibilia " must be regarded as mere 
subjective phantasms. A given table will present to one 
man a rectangular appearance, while to another it appears 
to have two acute angles and two obtuse angles ; to one 
man it appears brown, while to another, towards whom 
it reflects the light, it appears white and shiny. It is 
said, not wholly without plausibility, that these different 
shapes and different colours cannot co-exist simul 
taneously in the same place, and cannot therefore both 
be constituents of the physical world. This argument I 
must confess appeared to me until recently to be irre 
futable. The contrary opinion has, however, been ably 
maintained by Dr. T. P. Nunn in an article entitled : " Are 
Secondary Qualities Independent of Perception ? "* The 
supposed impossibility derives its apparent force from the 
phrase : "in the same place, and it is precisely in this 
phrase that its weakness lies. The conception of space 
is too often treated in philosophy even by those who on 
reflection would not defend such treatment as though it 
were as given, simple, and unambiguous as Kant, in his 
psychological innocence, supposed. It is the unperceived 
ambiguity of the word " place " which, as we shall shortly 
see, has caused the difficulties to realists and given an un 
deserved advantage to their opponents. Two " places " 
of different kinds are involved in every sense-datum, 
namely the place at which it appears and the place from 
which it appears. These belong to different spaces, 
although, as we shall see, it is possible, with certain 
limitations, to establish a correlation between them. 

1 Pvoc. /Irttf. Soc.. 1909-1910, pp. jQi-218. 



154 MYSTICISM AND LOGIC 

What we call the different appearances of the same thing 
to different observers are each in a space private to the 
observer concerned. No place in the private world of 
one observer is identical with a place in the private world 
of another observer. There is therefore no question of 
combining the different appearances in the one place ; 
and the fact that they cannot all exist in one place affords 
accordingly no ground whatever for questioning their 
physical reality. The " thing " of common sense may in 
fact be identified with the whole class of its appearances 
where, however, we must include among appearances 
not only those which are actual sense-data, but also 
those " sensibilia," if any, which, on grounds of con 
tinuity and resemblance, are to be regarded as belonging 
to the same system of appearances, although there 
happen to be no observers to whom they are data. 

An example may make this clearer. Suppose there are 
a number of people in a room, all seeing, as they say, the 
same tables and chairs, walls and pictures. No two of 
these people have exactly the same sense-data, yet there 
is sufficient similarity among their data to enable them 
to group together certain of these data as appearances of 
one " thing " to the several spectators, and others as 
appearances of another " thing." Besides the appear 
ances which a given thing in the room presents to the 
actual spectators, there are, we may suppose, other 
appearances which it would present to other possible 
spectators. If a man were to sit down between two 
others, the appearance which the room would present to 
him would be intermediate between the appearances 
which it presents to the two others : and although this 
appearance would not exist as it is without the sense 
organs, nerves and brain, of the newly arrived spectator, 
ttill it is not unnatural to suppose that, from the position 



SENSE-DATA AND PHYSICS 155 

which he now occupies, some appearance of the room 
existed before his arrival. This supposition, however, 
need merely be noticed and not insisted upon. 

Since the " thing " cannot, without indefensible par 
tiality, be identified with any single one of its appear 
ances, it came to be thought of as something distinct 
from all of them and underlying them. But by the prin 
ciple of Occam s razor, if the class of appearances will 
fulfil the purposes for the sake of which the thing was 
invented by the prehistoric metaphysicians to whom 
common sense is due, economy demands that we should 
identify the thing with the class of its appearances. It is 
not necessary to deny a substance or substratum underly 
ing these appearances ; it is merely expedient to abstain 
from asserting this unnecessary entity. Our procedure 
here is precisely analogous to that which has swept away 
from the philosophy of mathematics the useless menagerie 
of metaphysical monsters with which it used to be in 
fested. 

VI. CONSTRUCTIONS VERSUS INFERENCES 

Before proceeding to analyse and explain the am 
biguities of the word " place," a few general remarks on 
method are desirable. The supreme maxim in scientific 
philosophising is this : 

Wherever possible, logical constructions are to be sub 
stituted for inferred entities. 

Some examples of the substitution of construction for 
inference in the realm of mathematical philosophy may 
serve to elucidate the uses of this maxim. Take first the 
case of irrationals. In old days, irrationals were inferred 
as the supposed limits of series of rationals which had no 
rational limit ; but the objection to this procedure was 



156 MYSTICISM AND LOGIC 

that it left the existence of irrationals merely optative, 
and for this reason the stricter methods of the present 
day no longer tolerate such a definition. We now define 
an irrational number as a certain class of ratios, thus 
constructing it logically by means of ratios, instead of 
arriving at it by a doubtful inference from them. Take 
again the case of cardinal numbers. Two equally 
numerous collections appear to have something in 
common : this something is supposed to be their car 
dinal number. But so long as the cardinal number is 
inferred from the collections, not constructed in terms 
of them, its existence must remain in doubt, unless in 
virtue of a metaphysical postulate ad hoc. By defining 
the cardinal number of a given collection as the class of 
all equally numerous collections, we avoid the necessity 
of this metaphysical postulate, and thereby remove a 
needless element of doubt from the philosophy of arith 
metic. A similar method, as I have shown elsewhere, 
can be applied to classes themselves, which need not be 
supposed to have any metaphysical reality, but can be 
regarded as symbolically constructed fictions. 

The method by which the construction proceeds is 
closely analogous in these and all similar cases. Given a 
set of propositions nominally dealing with the supposed 
inferred entities, we observe the properties which are 
required of the supposed entities in order to make these 
propositions true. By dint of a little logical ingenuity, 
we then construct some logical function of less hypo 
thetical entities which has the requisite properties. This 
constructed function we substitute for the supposed in 
ferred entities, and thereby obtain a new and less doubtful 
interpretation of the body of propositions in question 
This method, so fruitful in the philosophy of mathematics, 
will be found equally applicable in the philosophy oi 



SENSE-DATA AND PHYSICS 157 

physics, where, I do not doubt, it would have been applied 
long ago but for the fact that all who have studied this 
subject hitherto have been completely ignorant of mathe 
matical logic. I myself cannot claim originality in the 
application of this method to physics, since I owe the 
suggestion and the stimulus for its application entirely 
to my friend and collaborator Dr. Whitehead, who is 
engaged in applying it to the more mathematical portions 
of the region intermediate between sense-data and the 
points, instants and particles of physics. 

A complete application of the method which substitutes j 
constructions for inferences would exhibit matter wholly / 
in terms of sense-data, and even, we may add, of the sense- V 
data of a single person, since the sense-data of others V 
cannot be known without some element of inference. \ 
This, however, must remain for the present an ideal, to s 
be approached as nearly as possible, but to be reached, if 
at all, only after a long preliminary labour of which as 
yet we can only see the very beginning. The inferences 
which are unavoidable can, however, be subjected to 
certain guiding principles. In the first place they should 
always be made perfectly explicit, and should be formulated 
in the most general manner possible. In the second place 
the inferred entities should, whenever this can be done, be 
similar to those whose existence is given, rather than, like 
the Kantian Ding an sich, something wholly remote from 
the data which nominally support the inference. The 
inferred entities which I shall allow myself are of two 
kinds : (a) the sense-data of other people, in favour of 
which there is the evidence of testimony, resting ulti 
mately upon the analogical argument in favour of minds 
other than my own ; (b) the " sensibilia " which would 
appear from places where there happen to be no minds, 
and which I suppose to be real although they are no one s 



158 MYSTICISM AND LOGIC 

data. Of these two classes of inferred entities, the first 
will probably be allowed to pass unchallenged. It would 
give me the greatest satisfaction to be able to dispense 
with it, and thus establish physics upon a solipsistic 
basis ; but those and I fear they are the majority in 
whom the human affections are stronger than the desire 
for logical economy, will, no doubt, not share my desire 
to render solipsism scientifically satisfactory. The second 
class of inferred entities raises much more serious ques 
tions. It may be thought monstrous to maintain that a 
thing can present any appearance at all in a place where 
no sense organs and nervous structure exist through which 
it could appear. I do not myself feel the monstrosity ; 
nevertheless I should regard these supposed appearances 
only in the light of a hypothetical scaffolding, to be used 
while the edifice of physics is being raised, though 
possibly capable of being removed as soon as the edifice is 
completed. These " sensibilia " which are not data to 
anyone are therefore to be taken rather as an illustrative 
hypothesis and as an aid in preliminary statement than 
as a dogmatic part of the philosophy of physics in its 
final form. 

VII. PRIVATE SPACE AND THE SPACE OF 
PERSPECTIVES 

We have now to explain the ambiguity in the word 
" place," and how it comes that two places of different 
sorts are associated with every sense-datum, namely the 
place at which it is and the place from which it is per 
ceived. The theory to be advocated is closely analogous 
to Leibniz s monadology, from which it differs chiefly in 
being less smooth and tidy. 

The first fact to notice is that, so far as can be dis 
covered, no sensibile is ever a datum to two people at 



SENSE-DATA AND PHYSICS 159 

once. The things seen by two different people are often 
closely similar, so similar that the same words can be used 
to denote them, without which communication with 
others concerning sensible objects would be impossible. 
But, in spite of this similarity, it would seem that some 
difference always arises from difference in the point of 
view. Thus each person, so far as his sense-data are con 
cerned, lives in a private world. This private world 
contains its own space, or rather spaces, for it would 
seem that only experience teaches us to correlate the 
space of sight with the space of touch and with the 
various other spaces of other senses. This multiplicity 
of private spaces, however, though interesting to the 
psychologist, is of no great importance in regard to our 
present problem, since a merely solipsistic experience 
enables us to correlate them into the one private space 
which embraces all our own sense-data. The place at 
which a sense-datum is, is a place in private space. This 
place therefore is different from any place in the private 
space of another percipient. For if we assume, as logical 
economy demands, that all position is relative, a place is 
only definable by the things in or around it, and therefore 
the same place cannot occur in two private worlds which 
have no common constituent. The question, therefore, 
of combining what we call different appearances of the 
same thing in the same place does not arise, and the fact 
that a given object appears to different spectators to 
have different shapes and colours affords no argument 
against the physical reality of all these shapes and 
colours. 

In addition to the private spaces belonging to the 
private worlds of different percipients, there is, however, 
another space, in which one whole private world counts 
as a. point, or at least as a spatial unit. This might be 



r6o MYSTICISM AND LOGIC 

described as the space of points oi view, since each 
private world may be regarded as the appearance 
which the universe presents from a certain point of 
view. I prefer, however, to speak of it as the space of 
perspectives, in order to obviate the suggestion that a 
private world is only real when someone views it. 
And for the same reason, when I wish to speak of a 
private world without assuming a percipient, I shall call 
it a " perspective." 

We have now to explain how the different perspectives 
are ordered in one space. This is effected by means of the 
correlated " sensibilia " which are regarded as the appear 
ances, in different perspectives, of one and the same thing. 
By moving, and by testimony, we discover that two 
different perspectives, though they cannot both contain 
the same " sensibilia," may nevertheless contain very 
similar ones ; and the spatial order of a certain group of 
"sensibilia" in a private space of one perspective is 
found to be identical with, or very similar to, the spatial 
order of the correlated " sensibilia " in the private space 
of another perspective. In this way one " sensibile " in 
one perspective is correlated with one " sensibile " in 
another. Such correlated " sensibilia " will be called 
" appearances of one thing." In Leibniz s monadology, 
since each monad mirrored the whole universe, there was 
in each perspective a " sensibile " which was an appear 
ance of each thing. In our system of perspectives, we 
make no such assumption of completeness. A given 
thing will have appearances in some perspectives, but 
presumably not in certain others. The " thing " being 
defined as the class of its appearances, if K is the class of 
perspectives in which a certain thing appears, then is 
a member of the multiplicative class of AC , K being a class 
ol mutually exclusive classes of " sensibilia." And 



SENSE-DATA AND PHYSICS 161 

similarly a perspective is a member of the multiplicative 
class of the things which appear in it. 

The arrangement of perspectives in a space is effected 
by means of the differences between the appearances of a 
given thing in the various perspectives. Suppose, say, 
that a certain penny appears in a number of different 
perspectives ; in some it looks larger and in some smaller, 
in some it looks circular, in others it presents the appear 
ance of an ellipse of varying eccentricity. We may collect 
together all those perspectives in which the appearance of 
the penny is circular. These we will place on one straight 
line, ordering them in a series by the variations in the 
apparent size of the penny. Those perspectives in which 
the penny appears as a straight line of a certain thickness 
will similarly be placed upon a plane (though in this case 
there will be many different perspectives in which the 
penny is of the same size ; when one arrangement is com 
pleted these will form a circle concentric with the penny) , 
and ordered as before by the apparent size of the penny. 
By such means, all those perspectives in which the penny 
presents a visual appearance can be arranged in a three- 
dimensional spatial order. Experience shows that the same 
spatial order of perspectives would have resulted if, instead 
of the penny, we had chosen any other thing which 
appeared in all the perspectives in question, or any other 
method of utilising the differences between the appearances 
of the same things in different perspectives. It is this 
empirical fact which has made it possible to construct 
the one all-embracing space of physics. 

The space whose construction has just been explained, 
and whose elements are whole perspectives, will be called 
" perspective-space." 



162 MYSTICISM AND LOGIC 

VIII. THE PLACING OF " THINGS " AND " SENSIBILIA " IN 
PERSPECTIVE SPACE 

The world which we have so far constructed is a world 
of six dimensions, since it is a three-dimensional series of 
perspectives, each of which is itself three-dimensional. 
We have now to explain the correlation between the per 
spective space and the various private spaces contained 
within the various perspectives severally. It is by means 
of this correlation that the one three-dimensional space 
of physics is constructed ; and it is because of the un 
conscious performance of this correlation that the dis 
tinction between perspective space and the percipient s 
private space has been blurred, with disastrous results 
for the philosophy of physics. Let us revert to our 
penny : the perspectives in which the penny appears 
larger are regarded as being nearer to the penny than 
those in which it appears smaller, but as far as experience 
goes the apparent size of the penny will not grow beyond 
a certain limit, namely, that where (as we say) the penny 
is so near the eye that if it were any nearer it could not 
be seen. By touch we may prolong the series until the 
penny touches the eye, but no further. If we have been 
travelling along a line of perspectives in the previously 
defined sense, we may, however, by imagining the penny 
removed, prolong the line of perspectives by means, say, 
of another penny ; and the same may be done with any 
other line of perspectives defined by means of the penny. 
All these lines meet in a certain place, that is, in a certain 
perspective. This perspective will be defined as " the 
place where the penny is." 

It is now evident in what sense two places in con 
structed physical space are associated with a given 
" sensibile." There is first the place which is the per- 



SENSE-DATA AND PHYSICS 163 

spective of which the " sensibile " is a member. This is 
the place from which the " sensibile " appears. Secondly 
there is the place where the thing is of which the " sen 
sibile " is a member, in other words an appearance ; this 
is the place at which the " sensibile " appears. The 
" sensibile " which is a member of one perspective is 
correlated with another perspective, namely, that which 
is the place where the thing is of which the " sensibile " 
is an appearance. To the psychologist the " place from 
which " is the more interesting, and the " sensibile " 
accordingly appears to him subjective and where the 
percipient is. To the physicist the " place at which " is 
the more interesting, and the " sensibile " accordingly 
appears to him physical and external. The causes, limits 
and partial justification of each of these two apparently 
incompatible views are evident from the above duplicity 
of places associated with a given " sensibile." 

We have seen that we can assign to a physical thing a 
place in the perspective space. In this way different 
parts of our body acquire positions in perspective space, 
and therefore there is a meaning (whether true or false 
need not much concern us) in saying that the perspective 
to which our sense-data belong is inside our head. Since 
our mind is correlated with the perspective to which our 
sense-data belong, we may regard this perspective as 
being the position of our mind in perspective space. If, 
therefore, this perspective is, in the above denned sense, 
inside our head, there is a good meaning for the state 
ment that the mind is in the head. We can now say of 
the various appearances of a given thing that some of 
them are nearer to the thing than others ; those are 
nearer which belong to perspectives that are nearer to 
" the place where the thing is." We can thus find a 
meaning, true or false, for the statement that more is to 



164 MYSTICISM AND LOGIC 

be learnt about a thing by examining it close to than by 
viewing it from a distance. We can also find a meaning 
for the phrase " the things which intervene between the 
subject and a thing of which an appearance is a datum 
to him." One reason often alleged for the subjectivity 
of sense-data is that the appearance of a thing may change 
when we find it hard to suppose that the thing itself has 
changed for example, when the change is due to our 
shutting our eyes, or to our screwing them up so as to 
make the thing look double. If the thing is defined as 
the class of its appearances (which is the definition adopted 
above), there is of course necessarily some change in the 
thing whenever any one of its appearances changes. 
Nevertheless there is a very important distinction between 
two different ways in which the appearances may change. 
If after looking at a thing I shut my eyes, the appearance 
of my eyes changes in every perspective in which there 
is such an appearance, whereas most of the appearances 
of the thing will remain unchanged. We may say, as a 
matter of definition, that a thing changes when, however 
near to the thing an appearance of it may be, there are 
changes in appearances as near as, or still nearer to, the 
thing. On the other hand we shall say that the change is 
in some other thing if all appearances of the thing which 
are at not more than a certain distance from the thing 
remain unchanged, while only comparatively distant 
appearances of the thing are altered. From this con 
sideration we are naturally led to the consideration of 
matter, which must be our next topic. 

IX. THE DEFINITION OF MATTER 

We defined the " physical thing " as the class of its 
appearances, but this can hardly be taken as a definition 
of matter. WP want to be able to express the fact that 



SENSE-DATA AND PHYSICS 165 

the appearance of a thing in a given perspective is 
causally affected by the matter between the thing and the 
perspective. We have found a meaning for " between a 
thing and a perspective." But we want matter to be 
something other than the whole class of appearances of a 
thing, in order to state the influence of matter on appear 
ances. 

We commonly assume that the information we get 
about a thing is more accurate when the thing is nearer. 
Far off, we see it is a man ; then we see it is Jones ; then 
we see he is smiling. Complete accuracy would only be 
attainable as a limit : if the appearances of Jones as we 
approach him tend towards a limit, that limit may be 
taken to be what Jones really is. It is obvious that from 
the point of view of physics the appearances of a thing 
close to " count " more than the appearances far off. We 
may therefore set up the following tentative definition : 

The matter of a given thing is the limit of its appear 
ances as their distance from the thing diminishes. 

It seems probable that there is something in this 
definition, but it is not quite satisfactory, because em 
pirically there is no such limit to be obtained from sense- 
data. The definition will have to be eked out by con 
structions and definitions. But probably it suggests the 
right direction in which to look. 

We are now in a position to understand in outline the 
reverse journey from matter to sense-data which is per 
formed by physics. The appearance of a thing in a given 
perspective is a function of the matter composing the 
thing and of the intervening matter. The appearance of 
a thing is altered by intervening smoke or mist, by blue 
spectacles or by alterations in the sense-organs or nerves 
of the percipient (which also must be reckoned as part of 
the intervening medium). The nearer we approach to 



166 MYSTICISM AND LOGIC 

the thing, the less its appearance is affected by the inter 
vening matter. As we travel further and further from the 
thing, its appearances diverge more and more froir their 
initial character ; and the causal laws of their divergence 
are to be stated in terms of the matter which lies between 
them and the thing. Since the appearances at very small 
distances are less affected by causes other than the thing 
itself, we come to think that the limit towards which these 
appearances tend as the distance diminishes is what the 
thing " really is," as opposed to what it merely seems to 
be. This, together with its necessity for the statement of 
causal laws, seems to be the source of the entirely erro 
neous feeling that matter is more " real " than sense- 
data. 

Consider for example the infinite divisibility of matter 
In looking at a given thing and approaching it, one sense- 
datum will become several, and each of these will again 
divide. Thus one appearance may represent many things, 
and to this process there seems no end. Hence in the 
limit, when we approach indefinitely near to the thing 
there will be an indefinite number of units of matte i 
corresponding to what, at a finite distance, is only one 
appearance. This is how infinite divisibility arises. 

The whole causal efficacy of a thing resides in its matter. 
This is in some sense an empirical fact, but it would be hard 
to state it precisely, because " causal efficacy " is difficult 
to define. 

What can be known empirically about the matter of a 
thing is only approximate, because we cannot get to know 
the appearances of the thing from very small distances, 
and cannot accurately infer the limit of these appearances. 
But it is inferred approximately by means of the appear 
ances we can observe. It then turns out that these 
appearances can be exhibited by physics as a function of 



SENSE-DATA AND PHYSICS 167 

the matter in our immediate neighbourhood ; e.g. the 
visual appearance of a distant object is a function of the 
light-waves that reach the eyes. This leads to confusions 
of thought, but offers no real difficulty. 

One appearance, of a visible object for example, is not 
sufficient to determine its other simultaneous appearances, 
although it goes a certain distance towards determining 
them. The determination of the hidden structure of a 
thing, so far as it is possible at all, can only be effected by 
means of elaborate dynamical inferences. 

X. TIME 1 

It seems that the one all-embracing time is a con 
struction, like the one all-embracing space. Physics 
itself has become conscious of this fact through the dis 
cussions connected with relativity. 

Between two perspectives which both belong to one 
person s experience, there will be a direct time-relation of 
before and after. This suggests a way of dividing history 
in the same sort of way as it is divided by different 
experiences, but without introducing experience or any 
thing mental : we may define a " biography " as every 
thing that is (directly) earlier or later than, or simul 
taneous with, a given " sensibile." This will give a series 
of perspectives, which might all form parts of one person s 
experience, though it is not necessary that all or any of 
them should actually do so. By this means, the history 
of the world is divided into a number of mutually exclusive 
biographies. 

1 On this subject, compare A Theory of Time and Space, by Mr. 
A. A. Robb (Camb. Univ. Press), which first suggested to me the views 
advocated here, though I have, for present purposes, omitted what is 
most interesting and novel in his theory. Mr. Robb has given a sketch 
of his theory in a pamphlet with the same title (Heffer and Sons. 
Cambridge. 1913). 



168 MYSTICISM AND LOGIC 

We have now to correlate the times in the different 
biographies. The natural thing would be to say that the 
appearances of a given (momentary) thing in two different 
perspectives belonging to different biographies are to be 
taken as simultaneous ; but this is not convenient. 
Suppose A shouts to B, and B replies as soon as he hears 
A s shout. Then between A s hearing of his own shout 
and his hearing of B s there is an interval ; thus if we 
made A s and B s hearing of the same shout exactly 
simultaneous with each other, we should have events 
exactly simultaneous with a given event but not with 
each other. To obviate this, we assume a " velocity of 
sound." That is, we assume that the time when B hears 
A s shout is half-way between the time when A hears his 
own shout and the time when he hears B s. In this way 
the correlation is effected. 

What has been said about sound applies of course 
equally to light. The general principle is that the 
appearances, in different perspectives, which are to be 
grouped together as constituting what a certain thing is 
at a certain moment, are not to be all regarded as being 
at that moment. On the contrary they spread outward 
from the thing with various velocities according to the 
nature of the appearances. Since no direct means exist 
of correlating the time in one biography with the time in 
another, this temporal grouping of the appearances 
belonging to a given thing at a given moment is in part 
conventional. Its motive is partly to secure the verifica 
tion of such maxims as that events which are exactly 
simultaneous with the same event are exactly simul 
taneous with one another, partly to secure convenience 
in the formulation of causal laws. 



SENSE-DATA AND PHYSICS 169 

XI. THE PERSISTENCE OF THINGS AND MATTER 

Apart from any of the fluctuating hypotheses of 
physics, three main problems arise in connecting the 
world of physics with the world of sense, namely : 

1. the construction of a single space ; 

2. the construction of a single time ; 

3. the construction of permanent things or matter. 

We have already considered the first and second of 
these problems ; it remains to consider the third. 

We have seen how correlated appearances in different 
perspectives are combined to form one " thing " at one 
moment in the all-embracing time of physics. We have 
now to consider how appearances at different times are 
combined as belonging to one " thing/ and how we 
arrive at the persistent " matter " of physics. The 
assumption of permanent substance, which technically 
underlies the procedure of physics, cannot of course be 
regarded as metaphysically legitimate : just as the one 
thing simultaneously seen by many people is a con 
struction, so the one thing seen at different times by the 
same or different people must be a construction, being in 
fact nothing but a certain grouping of certain " sensibilia." 

We have seen that the momentary state of a " thing " 
is an assemblage of " sensibilia," in different perspectives, 
not all simultaneous in the one constructed time, but 
spreading out from " the place where the thing is " with 
velocities depending upon the nature of the " sensibilia." 
The time at which the " thing " is in this state is the lower 
limit of the times at which these appearances occur. We 
have now to consider what leads us to speak of another 
set of appearances as belonging to the same " thing " at 
a different time. 



170 MYSTICISM AND LOGIC 

For this purpose, we may, at least to begin with, 
confine ourselves within a single biography. If we can 
always say when two " sensibilia " in a given biography 
are appearances of one thing, then, since we have seen 
how to connect " sensibilia " in different biographies as 
appearances of the same momentary state of a thing, we 
shall have all that is necessary for the complete con 
struction of the history of a thing. 

It is to be observed, to begin with, that the identity of 
a thing for common sense is not always correlated with 
the identity of matter for physics. A human body is one 
persisting thing for common sense, but for physics its 
matter is constantly changing. We may say, broadly, 
that the common-sense conception is based upon con 
tinuity in appearances at the ordinary distances of sense- 
data, while the physical conception is based upon the 
continuity of appearances at very small distances from 
the thing. It is probable that the common-sense con 
ception is not capable of complete precision. Let us there 
fore concentrate our attention upon the conception of the 
persistence of matter in physics. 

The first characteristic of two appearances of the same 
piece of matter at different times is continuity. The two 
appearances must be connected by a series of inter 
mediaries, which, if time and space form compact series, 
must themselves form a compact series. The colour of 
the leaves is difierent in autumn from what it is in summer; 
but we believe that the change occurs gradually, and that, 
if the colours are different at two given times, there are 
intermediate times at which the colours are intermediate 
between those at the given times. 

But there are two considerations that are important as 
regards continuity. 

First, it is largely hypothetical. We do not observe 



SENSE-DATA AND PHYSICS 171 

any one thing continuously, and it is merely a hypo 
thesis to assume that, while we are not observing it, it 
passes through conditions intermediate between those in 
which it is perceived. During uninterrupted observa 
tion, it is true, continuity is nearly verified ; but even 
here, when motions are very rapid, as in the case of 
explosions, the continuity is not actually capable of 
direct verification. Thus we can only say that the sense- 
data are found to permit a hypothetical complement of 
" sensibilia " such as will preserve continuity, and that 
therefore there may be such a complement. Since, how 
ever, we have already made such use of hypothetical 
" sensibilia," we will let this point pass, and admit such 
" sensibilia," as are required to preserve continuity. 

Secondly, continuity is not a sufficient criterion of 
material identity. It is true that in many cases, such as 
rocks, mountains, tables, chairs, etc., where the appear 
ances change slowly, continuity is sufficient, but in other 
cases, such as the parts of an approximately homogeneous 
fluid, it fails us utterly. We can travel by sensibly 
continuous gradations from any one drop of the sea at 
any one time to any other drop at any other time. We 
infer the motions of sea-water from the effects of the 
current, but they cannot be inferred from direct sensible 
observation together with the assumption of continuity. 

The characteristic required in addition to continuity is 
conformity with the laws of dynamics Starting from 
what common sense regards as persistent things, and 
making only such modifications as from time to time 
seem reasonable, we arrive at assemblages of " sensibilia " 
which are found to obey certain simple laws, namely those 
of dynamics. By regarding " sensibilia " at different 
times as belonging to the same piece of matter, we are 
able to define motion, which presupposes the assumption 



173 MYSTICISM AND LOGIC 

or construction of something persisting throughout the 
time of the motion. The motions which are regarded as 
occurring, during a period in which all the " sensibilia " 
and the times of their appearance are given, will be 
different according to the manner in which we combine 
" sensibilia " at different times as belonging to the same 
piece of matter. Thus even when the whole history of 
the world is given in every particular, the question what 
motions take place is still to a certain extent arbitrary 
even after the assumption of continuity. Experience 
shows that it is possible to determine motions in such a 
way as to satisfy the laws of dynamics, and that this 
determination, roughly and on the whole, is fairly in 
agreement with the common-sense opinions about per 
sistent things. This determination, therefore, is adopted, 
and leads to a criterion by which we can determine, some 
times practically, sometimes only theoretically, whether 
two appearances at different times are to be regarded as 
belonging to the same piece of matter. The persistence 
of all matter throughout all time can, I imagine, be 
secured by definition. 

To recommend this conclusion, we must consider what 
it is that is proved by the empirical success of physics. 
What is proved is that its hypotheses, though unverifiable 
where they go beyond sense-data, are at no point in 
contradiction with sense-data, but, on the contrary, are 
ideally such as to render all sense-data calculable when a 
sufficient collection of " sensibilia " is given. Now 
physics has found it empirically possible to collect sense- 
data into series, each series being regarded as belonging 
to one " thing," and behaving, with regard to the laws 
of physics, in a way in which series not belonging to one 
thing would in general not behave. If it is to be un 
ambiguous whether two appearances belong to the same 



SENSE-DATA AND PHYSICS 173 

thing or not, there must be only one way of grouping 
appearances so that the resulting things obey the laws of 
physics. It would be very difficult to prove that this is 
the case, but for our present purposes we may let this 
point pass, and assume that there is only one way. Thus 
we may lay down the following definition : Physical 
things are those series of appearances whose matter obeys 
the laws of physics. That such series exist is an empirical 
fact, which constitutes the verifiability of physics. 



XII. ILLUSIONS, HALLUCINATIONS, AND DREAMS 

It remains to ask how, in our system, we are to find a 
place for sense-data which apparently fail to have the 
usual connection with the world of physics. Such sense- 
data are of various kinds, requiring somewhat different 
treatment. But all are of the sort that would be called 
" unreal," and therefore, before embarking upon the dis 
cussion, certain logical remarks must be made upon the 
conceptions of reality and unreality. 

Mr. A. Wolf 1 says : 

" The conception of mind as a system of transparent 
activities is, I think, also untenable because of its failure 
to account for the very possibility of dreams and hallu 
cinations. It seems impossible to realise how a bare, 
transparent activity can be directed to what is not there, 
to apprehend what is not given." 

This statement is one which, probably, most people 
would endorse. But it is open to two objections. First 
it is difficult to see how an activity, however un- " trans 
parent," can be directed towards a nothing : a term of a 
relation cannot be a mere nonentity. Secondly, no reason 

1 " Natural Realism and Present Tendencies in Philosophy," Proc 
Arist. Sac., 1908-1909, p. 165. 



174 MYSTICISM AND LOGIC 

is given, and I am convinced that none can be given, for 
the assertion that dream-objects are not " there " and 
not " given." Let us take the second point first 

(1) The belief that dream-objects are not given comes, 
I think, from failure to distinguish, as regards waking 
life, between the sense-datum and the corresponding 
" thing/ In dreams, there is no such corresponding 
" thing " as the dreamer supposes ; if, therefore, the 
" thing " were given in waking life, as e.g. Meinong 
maintains, 1 then there would be a difference in respect of 
givenness between dreams and waking life. But if, as 
we have maintained, what is given is never the thing, but 
merely one of the " sensibilia " which compose the thing, 
then what we apprehend in a dream is just as much given 
as what we apprehend in waking life. 

Exactly the same argument applies as to the dream- 
objects being " there." They have their position in the 
private space of the perspective of the dreamer ; where 
they fail is in their correlation with other private spaces 
and therefore with perspective space. But in the only 
sense in which " there " can be a datum, they are " there " 
just as truly as any of the sense-data of waking life. 

(2) The conception of " illusion " or " unreality," and 
the correlative conception of " reality," are generally 
used in a way which embodies profound logical con 
fusions. Words that go in pairs, such as " real " and 
"unreal," "existent" and "non-existent," "valid" 
and " invalid," etc., are all derived from the one funda 
mental pair, "true" and "false." Now "true" and 
" false " are applicable only except in derivative signifi 
cations to propositions. Thus wherever the above pairs 
can be significantly applied, we must be dealing either 
with propositions or with such incomplete phrases as 

1 Die Erfahrungsgrundlagen unseres Wissens, p. 28. 



SENSE-DAiA AND PHYSICS 175 

only acquire meaning when put into a context which, 
with them, forms a proposition. Thus such pairs of words 
can be applied to descriptions, 1 but not to proper names : 
in other words, they have no application whatever to 
data, but only to entities or non-entities described in 
terms of data. 

Let us illustrate by the terms " existence " and " non- 
existence." Given any datum x, it is meaningless either 
to assert or to deny that x " exists." We might be 
tempted to say : " Of course x exists, for otherwise it 
could not be a datum." But such a statement is really 
meaningless, although it is significant and true to say 
" My present sense-datum exists," and it may also be 
true that " x is my present sense-datum." The inference 
from these two propositions to " x exists " is one which 
seems irresistible to people unaccustomed to logic ; yet 
the apparent proposition inferred is not merely false, but 
strictly meaningless. To say " My present sense-datum 
exists " is to say (roughly) : " There is an object of which 
my present sense-datum is a description." But we 
cannot say : " There is an object of which x is a 
description," because x is (in the case we are supposing) 
a name, not a description. Dr. Whitehead and I have 
explained this point fully elsewhere (loc. cit.) with the 
help of symbols, without which it is hard to understand ; 
I shall not therefore here repeat the demonstration of the 
above propositions, but shall proceed with their applica 
tion to our present problem. 

The fact that " existence " is only applicable to 
descriptions is concealed by the use of what are gram 
matically proper names in a way which really transforms 
them into descriptions. It is, for example, a legitimate 

1 Cf. Principia Mathematica, Vol. I, * 14, and Introduction, Chap. 
III. For the definition of ixittencr.. rf. * 14. 02. 



176 MYSTICISM AND LOGIC 

question whether Homer existed ; but here " Homer " 
means " the author of the Homeric poems/ and is a 
description. Similarly we may ask whether God exists ; 
but then " God " means " the Supreme Being " or " the 
ens realissimum " or whatever other description we may 
preler. If " God " were a proper name, God would have 
to be a datum ; and then no question could arise as to 
His existence. The distinction between existence and 
other predicates, which Kant obscurely felt, is brought 
to light by the theory of descriptions, and is seen to 
remove " existence " altogether from the fundamental 
notions of metaphysics. 

What has been said about " existence " applies equally 
to " reality," which may, in fact, be taken as synonymous 
with " existence." Concerning the immediate objects in 
illusions, hallucinations, and dreams, it is meaningless to 
ask whether they " exist " or are " real." There they are, 
and that ends the matter. But we may legitimately 
inquire as to the existence or reality of " things " or other 
" sensibilia " inferred from such objects. It is the un 
reality of these " things " and other " sensibilia," together 
with a failure to notice that they are not data, which has 
led to the view that the objects of dreams are unreal. 

We may now apply these considerations in detail to the 
stock arguments against realism, though what is to be said 
will be mainly a repetition of what others have said before. 

d) We have first the variety of normal appearances, 
supposed to be incompatible. This is the case of the 
different shapes and colours which a given thing presents 
to different spectators. Locke s water which seems both 
hot and cold belongs to this class of cases. Our system 
of different perspectives fully accounts for these cases, 
and shows that they afford no argument against realism. 

(2) We have cases where the correlation between 



SENSE-DATA AND PHYSICS 177 

different senses is unusual. The bent stick in water 
belongs here. People say it looks bent but is straight : 
this only means that it is straight to the touch, though 
bent to sight. There is no " illusion/ but only a false 
inference, if we think that the stick would feel bent to 
the touch. The stick would look just as bent in a photo 
graph, and, as Mr. Gladstone used to say, " the photo 
graph cannot lie." 1 The case of seeing double also 
belongs here, though in this case the cause of the unusual 
correlation is physiological, and would therefore not 
operate in a photograph. It is a mistake to ask whether 
the " thing " is duplicated when we see it double. The 
" thing " is a whole system of " sensibilia," and it is only 
those visual " sensibilia " which are data to the per 
cipient that are duplicated. The phenomenon has a 
purely physiological explanation ; indeed, in view of our 
having two eyes, it is in less need of explanation than the 
single visual sense-datum which we normally obtain from 
the things on which we focus. 

(3) We come now to cases like dreams, which may, at 
the moment of dreaming, contain nothing to arouse sus 
picion, but are condemned on the ground of their supposed 
incompatibility with earlier and later data. Of course it 
often happens that dream-objects fail to behave in the 
accustomed manner : heavy objects fly, solid objects melt, 
babies turn into pigs or undergo even greater changes. 
But none of these unusual occurrences need happen in a 
dream, and it is not on account of such occurrences that 
dream-objects are called " unreal." It is their lack ol 
continuity with the dreamer s past and future that makes 
him, when he wakes, condemn them ; and it is their lack 

1 Cf. Edwin B. Holt, The Place of Illusory Experience in a Realistic 
World. " The New Kealism," p. 30^, both on this point and as 
double. 



178 MYSTICISM AND LOGIC 

of correlation with other private worlds that makes 
others condemn them. Omitting the latter ground, our 
reason for condemning them is that the " things " which 
we infer from them cannot be combined according to the 
laws of physics with the " things " inferred from waking 
sense-data. This might be used to condemn the " things " 
inferred from the data of dreams. Dream-data are no 
doubt appearances of " things," but not of such " things " 
as the dreamer supposes. I have no wish to combat 
psychological theories of dreams, such as those of the 
psycho-analysts. But there certainly are cases where 
(whatever psychological causes may contribute) the 
presence of physical causes also is very evident. For 
instance, a door banging may produce a dream of a naval 
engagement, with images of battleships and sea and smoke. 
The whole dream will be an appearance of the door bang 
ing, but owing to the peculiar condition of the body 
(especially the brain) during sleep, this appearance is not 
that expected to be produced by a door banging, and thus 
the dreamer is led to entertain false beliefs. But his 
sense-data are still physical, and are such as a completed 
physics would include and calculate. 

(4) The last class of illusions are those which 
cannot be discovered within one person s experience, 
except through the discovery of discrepancies with 
the experiences of others. Dreams might conceivably 
belong to this class, if they were jointed sufficiently 
neatly into waking life ; but the chief instances 
are recurrent sensory hallucinations of the kind 
that lead to insanity. What makes the patient, in such 
cases, become what others call insane is the fact that, 
within his own experience, there is nothing to show that 
the hallucinatory sense-data do not have the usual kind 
of connection with " sensibilia " in other perspectives. 
Of course he may learn this through testimony, but he 



SENSE-DATA AND PHYSICS 179 

probably finds it simpler to suppose that the testimony is 
untrue and that he is being wilfully deceived. There is, 
so far as I can see, no theoretical criterion by which the 
patient can decide, in such a case, between the two 
equally satisfactory hypotheses of his madness and of his 
friends mendacity. 

From the above instances it would appear that ab 
normal sense-data, of the kind which we regard as decep 
tive, have intrinsically just the same status as any others, 
but differ as regards their correlations or causal connec 
tions with other " sensibilia " and with " things." Since 
the usual correlations and connections become part of 
our unreflective expectations, and even seem, except to 
the psychologist, to form part of our data, it comes to be 
thought, mistakenly, that in such cases the data are un 
real, whereas they are merely the causes of false infer 
ences. The fact that correlations and connections of un 
usual kinds occur adds to the difficulty of inferring things 
from sense and of expressing physics in terms of sense- 
data. But the unusualness would seem to be always 
physically or physiologically explicable, and therefore 
raises only a complication, not a philosophical objection. 

I conclude, therefore, that no valid objection exists to 
the view which regards sense-data as part of the actual 
substance of the physical world, and that, on the other 
hand, this view is the only one which accounts for the 
empirical verifiability of physics. In the present paper, 
I have given only a rough preliminary sketch. In par 
ticular, the part played by time in the construction of the 
physical world is, I think, more fundamental than would 
appear from the above account. 1 should hope that, 
with further elaboration, the part played by unper- 
ceived " sensibib a " could be indefinitely diminished, 
probably by invoking the history of a " thing " to eke out 
the inferences derivable from its momentary appearance. 



DC 

ON THE NOTION OF CAUSE 

T N the following paper I wish, first, to maintain that 
-- the word " cause " is so inextricably bound up with 
misleading associations as to make its complete extrusion 
from the philosophical vocabulary desirable ; secondly, 
to inquire what principle, if any, is employed in science 
in place of the supposed " law of causality " which philo 
sophers imagine to be employed ; thirdly, to exhibit 
certain confusions, especially in regard to teleology and 
determinism, which appear to me to be connected with 
erroneous notions as to causality. 

All philosophers, of every school, imagine that causa 
tion is one of the fundamental axioms or postulates of 
science, yet, oddly enough, in advanced sciences such as 
gravitational astronomy, the word " cause " never occurs. 
Dr. James Ward, in his Naturalism and Agnosticism, 
makes this a ground of complaint against physics : the 
business of those who wish to ascertain the ultimate truth 
about the world, he apparently thinks, should be the 
discovery of causes, yet physics never even seeks them. 
To me it seems that philosophy ought not to assume such 
legislative functions, and that the reason why physics 
has ceased to look for causes is that, in fact, there are no 
such things. The law of causality, I believe, like much 
that passes muster among philosophers, is a relic of a 
bygone age, surviving, like the monarchy, onlv because 
it is erroneously supposed to do no harm. 



ON THE NOTION OF CAUSE 181 

In order to find out what philosophers commonly 
understand by " cause," I consulted Baldwin s Dictionary, 
and was rewarded beyond my expectations, for I found 
the following three mutually incompatible definitions : 

" CAUSALITY, (i) The necessary connection of events 
in the time-series. . . . 

" CAUSE (notion of). Whatever may be included in 
the thought or perception of a process as taking 
place in consequence of another process. . . . 

" CAUSE AND EFFECT, (i) Cause and effect ... are 
correlative terms denoting any two distinguish 
able things, phases, or aspects of reality, which 
are so related to each other that whenever the 
first ceases to exist the second comes into exist 
ence immediately after, and whenever the second 
comes into existence the first has ceased to exist 
immediately before/ 

Let us consider these three definitions in turn. The 
first, obviously, is unintelligible without a definition of 
" necessary." Under this head, Baldwin s Dictionary 
gives the following : 

" NECESSARY. That is necessary which not only is 
true, but would be true under all circumstances. 
Something more than brute compulsion is, there 
fore, involved in the conception ; there is a 
general law under which the thing takes place." 

The notion of cause is so intimately connected with 
that of necessity that it will be no digression to linger 
over the above definition, with a view to discovering, if 
possible, some meaning of which it is capable ; for, as it 
stands, it is very far from having any definite signification. 

The first point to notice is that, if any meaning is to he 
given to the phrase " would be true under all circum 
stances," the subject of it must be a pro positional func- 

M 



182 MYSTICISM AND LOGIC 

tion, not a proposition. 1 A proposition is simply true or 
false, and that ends the matter : there can be no ques 
tion of " circumstances." " Charles I s head was cut off " 
is just as true in summer as in winter, on Sundays as on 
Mondays. Thus when it is worth saying that something 
"would be true under all circumstances," the something 
in question must be a prepositional function, i.e. an 
expression containing a variable, and becoming a pro 
position when a value is assigned to the variable ; the 
varying " circumstances " alluded to are then the 
different values of which the variable is capable. Thus if 
" necessary " means " what is true under all circum 
stances," then " if % is a man, x is mortal " is necessary, 
because it is true for any possible value of x. Thus we 
should be led to the following definition : 

" NECESSARY is a predicate of a propositional function, 
meaning that it is true for all possible values of 
its argument or arguments." 

Unfortunately, however, the definition in Baldwin s 
Dictionary says that what is necessary is not only " true 
under all circumstances " Bjit is also " true." Now these 
two are incompatible. Only propositions can be " true," 
and only propositional functions can be " true under all 
circumstances " Hence the definition as it stands is 
nonsense. What is meant seems to be this : "A pro 
position is necessary when it is a value of a propositional 
function which is true under all circumstances, i.e. for all 
values of its argument or arguments." But if we adopt 
this definition, the same proposition will be necessary or 
contingent according as we choose one or other of its 

1 A propositional function is an expression containing a variable, 01 
undetermined constituent, "and becoming a proposition as soon as a 
definite value is assigned to the variable. Examples are : " A is A," 
" * is a number." The variable is called the argument of the function. 



ON THE NOTION OF CAUSE 183 

terms as the argument to our prepositional function. For 
example, "if Socrates is a man, Socrates is mortal," is 
necessary if Socrates is chosen as argument, but not if 
man or mortal is chosen. Again, " if Socrates is a man, 
Plato is mortal," will be necessary if either Socrates or 
man is chosen as argument, but not if Plato or mortal is 
chosen. However, this difficulty can be overcome by 
specifying the constituent which is to be regarded as 
argument, and we thus arrive at the following definition : 

" A proposition is necessary with respect to a given 
constituent if it remains true when that constituent is 
altered in any way compatible with the proposition re 
maining significant." 

We may now apply this definition to the definition of 
causality quoted above. It is obvious that the argument 
must be the time at which the earlier event occurs. Thus 
an instance of causality will be such as : "If the event 
e l occurs at the time t lt it will be followed by the event 
e t ." This proposition is intended to be necessary with 
respect to t lt i.e. to remain true however t l may be 
varied. Causality, as a universal law, will then be the 
following : I" Given any event e lt there is an event e z 
such that, whenever e l occurs, e^ occurs later."] But 
before this can be considered precise, we must specify 
how much later e t is to occur. Thus the principle be 
comes : 

( Given any event e lt there is an event 2 and a time- 
interval T such that, whenever e l occurs, d 2 follows after 
an interval r . J 

I am not concerned as yet to consider whether this law 
is true or false. For the present, I am merely concerned 
to discover what the law of causality is supposed to be. 
I pass, therefore, to the other definitions quoted above. 



184 MYSTICISM AND LOGIC 

The second definition need not detain us long, for two 
reasons. First, because it is psychological : not the 
" thought or perception " of a process, but the process 
itself, must be what concerns us in considering causality. 
Secondly, because it is circular : in speaking of a process 
as " taking place in consequence of " another process, it 
introduces the very notion of cause which was to be 
denned. 

The third definition is by far the most precise ; indeed 
as regards clearness it leaves nothing to be desired. But 
a great difficulty is caused by the temporal contiguity of 
cause and effect which the definition asserts. No two 
instants are contiguous, since the time-series is compact ; 
hence either the cause or the effect or both must, if the 
definition is correct, endure for a finite time ; indeed, by 
the wording of the definition it is plain^that both are 
assumed to endure for a finite time. But then we are 
faced with a dilemma : if the cause is a process involving 
change within itself, we shall require (if causality is uni 
versal) causal relations between its earlier and later parts ; 
moreover, it would seem that only the later parts can be 
relevant to the effect, since the earlier parts are not 
contiguous to the effect, and therefore (by the definition) 
cannot influence the effect. Thus we shall be led to 
diminish the duration of the cause without limit, and 
however much we may diminish it, there will still 
remain an earlier part which might be altered without 
altering the effect, so that the true cause, as defined, will 
not have been reached, for it will be obsejved that the 
definition excludes plurality of causes. If, on the other 
hand, the cause is purely static, involving no change 
within itself, then, in the first place, no such cause is to 
be found in nature, and -in the second place, it seems 
strange too strange to be accepted, in spite of bare 



ON THE NOTION OF CAUSE 185 

logical possibility that the cause, after existing placidly 
for some time, should suddenly explode into the effect, 
when it might just as well have done so at any earlier 
time, or have gone on unchanged without producing its 
effect. This dilemma, therefore, is fatal to the view that 
cause and effect can be contiguous in time ; if there are 
causes and effects, they must be separated by a finite 
time-interval r, as was assumed in the above inter 
pretation of the first definition. 

What is essentially the same statement of the law of 
causality as the one elicited above from the first of 
Baldwin s definitions is given by other philosophers. 
Thus John Stuart Mill says : 

" The Law of Causation, the recognition of which is the 
main pillar of inductive science, is but the familiar truth, 
that x in variability of succession is found by observation 
to. obtain between every fact in nature and some other 
fact which has preceded it." 1 

And Bergson, who has rightly perceived that the law 
as stated by philosophers is worthless, nevertheless con 
tinues to suppose that it is used in science. Thus he 
says : 

" Now, it is argued, this law [the law of causality] 
means that every phenomenon is determined by its 
conditions, or, in other words, that the same causes 
produce the same effects. 

And again - 

" We perceive physical phenomena, and these pheno 
mena obey laws. This means : (i) That phenomena 
a, b, c, d, previously perceived, can occur again in the 
same shape ; (2) that a certain phenomenon P, which 

1 Logic, Bk. Ill, Chap. V, 2. 
7 Time and Free- Will, p. JQQ. 



186 MYSTICISM AND LOGIC 

appeared after the conditions a, b, c, d, and after these 
conditions only, will not fail to recur as soon as the same 
conditions are again present." 1 

A great part of Bergson s attack on science rests on the 
assumption that it employs this principle. In fact, it 
employs no such principle, but philosophers even 
Bergson are too apt to take their views on science from 
each other, not from science. As to what the principle 
is, there is a fair consensus among philosophers of different 
schools. There are, however, a number of difficulties 
which at once arise. I omit the question of plurality of 
causes for the present, since other graver questions have 
to be considered. Two of these, which are forced on our 
attention by the above statement of the law, are the 
foil owing : 

(1) What is meant by an " event " ? 

(2) How long may the time-interval be between cause 

and effect ? 

(i) An "event," in the statement of the law, is ob- 
viously intended to be something that is likely to recur 
since otherwise the law becomes trivial. It follows that 
an " event " is not a particular, but some_unjy^;sjji of 
which there may be many instances. It follows also that 
an " event " must be something short of the whole state 
of the universe, since it is highly improbable that this will 
recur. What is meant by an " event " is something like 
striking a match, or dropping a penny into the slot of an 
automatic machine. If such an event is to recur, it must 
not be denned too narrowly : we must not state with 
what degree of force the match is to be struck, nor what 
is to be the temperature of the penny. For if such con 
siderations were relevant, our " event " would occur at 

1 Ttmr and Ftee Will. p. 202. 



ON THE NOTION OF CAUSE 187 

most once, and the law would cease to give information. 
An " event," then, is a universal defined sufficiently 
widely to admit of many particular occurrences in time 
beingjnstaiic.es of it. 

(2) The next question concerns the time-interval. 
Philosophers, no doubt, think of cause and effect as 
contiguous in time, but this, for reasons already given, is 
impossible. Hence, since there are no infinitesimal time- 
intervals, there must be some finite lapse of time r 
between cause and effect. This, however, at once raises 
insuperable difficulties. However short we make the 
interval T, something may happen during this interval 
which prevents the expected result. I put my penny in 
the slot, but before I can draw out my ticket there is an 
earthquake which upsets the machine and my calcula 
tions. In order to be sure of the expected effect, we 
must know that there is nothing in the environment to 
interfere with it. But this means that the supposed 
cause is not, by itself, adequate to insure the effect. 
And as soon as we include the environment, the prob 
ability of repetition is diminished, until at last, when the 
whole environment is included, the probability of repeti 
tion be.comes almost nil. 

In spite of these difficulties, it must, of course, be 
admitted that many fairly dependable regularities of 
sequence occur in daily life. It is these! regularities that 
have suggested the supposed law of causality ; where they 
are found to fail, it is thought that a better formulation 
could have been found which would have never failed. 
I am far from denying that there may be such sequences 
which in fact never do fail. It may be that there will 
never be an exception to the rule that when a stone of 
more than a certain mass, moving with more than a 
certain velocity, comes in contact with a pane of glass of 



188 MYSTICISM AND LOGIC 

less than a certain thickness, the glass breaks. I also do 
not deny that the observation of such regularities, even 
when they are not without exceptions, is useful in the 
infancy of a science : the observation that unsupported 
bodies in air usually fall was a stage on the way to the 
law of gravitation. What I deny is that science assumes 
the existence of invariable uniformities of sequence of 
this kind, or that it aims at discovering them. All such 
uniformities, as we saw, depend upon a certain vagueness 
in the definition of the " events." That bodies fall is a 
vague qualitative statement ; science wishes to know 
how fast they fall. This depends upon the shape of the 
bodies and the density of the air. It is true that there is 
more nearly uniformity when they fall in a vacuum ; so 
far as Galileo could observe, the uniformity is then com 
plete. But later it appeared that even there the latitude 
made a difference, and the altitude. Theoretically, the 
position of the sun and moon must make a difference. 
In short, every advance in a science takes us farther 
away from the crude uniformities which are first observed, 
into greater differentiation of antecedent and consequent, 
and into a continually wider circle of antecedents recog 
nised as relevant.^. 

The principle "same cause, same effect, 1 which philo 
sophers imagine to be vital to science, is therefore utterly 
otiose. As soon as the antecedents have been given 
sufficiently fully to enable the consequent to be calcu 
lated with some exactitude, the antecedents have be 
come so complicated that it is very unlikely they will 
ever recur. Hence, if this were the principle involved, 
science would remain utterly sterile. 

The importance of these considerations lies partly in 
the fact that they le^,d to a more correct account of 
scientific procedure* partly in the fact that they remove 



ON THE NOTION OF CAUSE 189 

the analogy with human volition which makes the con 
ception of cause such a fruitful source of fallacies. The 
latter point will become clearer by the help of some 
illustrations. For this purpose I shall consider a few 
maxims which have played a great part in the history of 
philosophy. 

(1) f Cause and effect must more or less resemble each 
other, j This principle was prominent in the philosophy 
of occasionalism, and is still by no means extinct. It is 
still often thought, for example, that mind could not 
have grown up in a universe which previously contained 
nothing mental, and one ground for this belief is that 
matter is too dissimilar from mind to have been able to 
cause it. Or, more particularly, what are termed the 
nobler parts of our nature are supposed to be inexplicable, 
unless the universe always contained something at least 
equally noble which could cause them. All such views 
seem to depend upon assuming some unduly simplified 
law of causality ; for, in any legitimate sense of " cause " 
and " effect," science seems to show that they are 
usually very widely dissimilar, the " cause " being, in 
fact, two states of the whole universe, and the " effect " 
some particular event. 

(2) ^Cause is analogous to volition, since there must 
be an intelligible nexus between cause and effect, j This 
maxim is, I think, often unconsciously in the imagina 
tions of philosophers who would reject it when explicitly 
stated. It is probably operative in the view we have 
just been considering, that mind could not have resulted 
from a purely material world. I do not profess to know 
what is meant by " intelligible " ; it seems to mean 
" familiar to imagination." Nothing is less " intelli 
gible," in any other sense, than the connection between 



MYSTICISM AND LOGIC 

an act of will and its fulfilment. But obviously the sort 
of nexus desired between cause and effect is such as could 
only hold between the " events " which the supposed 
law of causality contemplates ; the laws which replace 
causality in such a science as physics leave no room for 
any two. events between which a nexus could be sought. 

(3) |" The cause compels the effect in some sense in 
which the effect does not compel the cause/ This belief 
seems largely operative in the dislike of determinism ; 
but, as a matter of fact, it is connected with our second 
maxim, and falls as soon as that is abandoned. We may 
define " compulsion " as follows : " Any set of circum 
stances is said to compel A when A desires to do some 
thing which the circumstances prevent, or to abstain 
from something which the circumstances cause." This 
presupposes that some meaning has been found for the 
word " cause " a point to which I shall return later. 
What I want to make clear at present is that compulsion 
is a very complex notion, involving thwarted desire. So 
long as a person does what he wishes to do, there is no 
compulsion, however much his wishes may be calculable 
by the help of earlier events. And where desire does not 
come in, there can be no question of compulsion. Hence 
it is, in general, misleading to regard the cause as com 
pelling the effect, 

A vaguer form of the same maxim substitutes the word 
" determine " for the word " compel " ; we are told that 
the cause determines the effect in a sense in which the 
effect does not determine the cause. It is not quite clear 
what is meant by " determining " ; the only precise 
sense, so far as I know, is that of a function or one-many 
relation. If we admit plurality of causes, but not of 
effects, that is, if we suppose that, given the cause, the 
effect must be such and such, but, given the effect, the 



ON THE NOTION OF CAUSE 191 

cause may have been one of many alternatives, then we 
may say that the cause determines the effect, but not the 
effect the cause. Plurality of causes, however, results 
only from conceiving the effect vaguely and narrowly 
and the cause precisely and widely. Many antecedents 
may " cause " a man s death, because his death is vague 
and narrow. But if we adopt the opposite course, taking 
as the " cause " the drinking of a dose of arsenic, and as 
the " effect " the whole state of the world five minutes 
later, we shall have plurality of effects instead of plurality 
of causes. Thus the supposed lack of symmetry between 
" causej and " effect " is illusory. 

(4) f^A cause cannot operate when it has ceased to 
exist, because what has ceased to exist is nothing. j This 
is a common maxim, and a still more common unex 
pressed prejudice. It has, I fancy, a good deal to do 
with the attractiveness of Bergson s " duree " : since the 
past has effects now, it must still exist in some sense. 
The mistake in this maxim consists in the supposition 
that causes " operate " at all. A volition " operates " 
when what it wills takes place ; but nothing can operate 
except a volition. The belief that causes " operate " 
results from assimilating them, consciously or uncon 
sciously, to volitions. We have already seen that, if 
there are causes at all, they must be separated by a finite 
interval of time from their effects, and thus cause their 
effects after they have ceased to exist. 

It may be objected to the above definition of a volition 
" operating " that it only operates when it " causes " 
what it wills, not when it merely happens to be followed 
by what it wills. This certainly represents the usual view 
of what is meant by a volition " operating," but as it 
involves the very view of causation which we are engaged 
in combating, it is not open to us as a definition. We 



I 9 2 MYSTICISM AND LOGIC 

may say that a volition "operates" when there is some 
law in virtue of which a similar volition in rather similar 
circumstances will usually be followed by what it wills. 
But this is a vague conception, and introduces ideas 
which we have not yet considered. What is chiefly im 
portant to notice is that the usual notion of " operating " 
is not open to us if we reject, as I contend that we should, 
the usual notion of causation. 

(5) " A cause cannot operate except where it is." This 
maxim is very widespread ; it was urged against Newton, 
and has remained a source of prejudice against " action at 
a distance." In philosophy it has led to a denial of 
transient action, and thence to monism or Leibnizian 
monadism. Like the analogous maxim concerning tem 
poral contiguity, it rests upon the assumption that causes 
" operate," i.e. that they are in some obscure way 
analogous to volitions. And, as in the case of temporal 
contiguity, the inferences drawn from this maxim are 
wholly groundless. 

I return now to the question, What law or laws 
can be found to take the place of the supposed law of 
causality ? 

First, without passing beyond such uniformities of 
sequence as are contemplated by the traditional law, we 
may admit that, if any such sequence has been observed 
in a great many cases, and has never been found to fail, 
there is an inductive probability that it will be found to 
hold in future cases. If stones have hitherto been found 
to break windows, it is probable that they will continue 
to do so. This, of course, assumes the inductive principle, 
of which the truth may reasonably be questioned ; but 
as this principle is not our present concern, I shall in this 
discussion treat it as indubitable We may then say, in 
the case of any siich frequently observed sequence, that 



ON THE NOTION OF CAUSE 193 

the earlier event is the cause and the later event the 
effect. 

Several considerations, however, make such special 
sequences very different from the traditional relation of 
cause and effect. In the first place, the sequence, in any 
hitherto unobserved instance, is no more than probable, 
whereas the relation of cause and effect was supposed to 
be necessary. I do not mean by this merely that we are 
not sure of having discovered a true case of cause and 
effect ; I mean that, even when we have a case of cause 
and effect in our present sense, all that is meant is that 
on grounds of observation, it is probable that when one 
occurs the other will also occur. Thus in our present 
sense, A may be the cause of B even if there actually are 
cases where B does not follow A. Striking a match will 
be the cause of its igniting, in spite of the fact that some 
matches are damp and fail to ignite. 

In the second place, it will not be assumed that every 
event has some antecedent which is its cause in this 
sense ; we shall only believe in causal sequences where 
we find them, without any presumption that they always 
are to be found. 

In the third place, any case of sufficiently frequent 
sequence will be causal in our present sense ; for example, 
we shall not refuse to say that night is the cause of day. 
Our repugnance to saying this arises from the ease with 
w iicii we can imagine the sequence to fail, but owing to 
the fact that cause and effect must be separated by a 
finite interval of time, any such sequence might fail 
through the interposition of other circumstances in the 
interval. Mill, discussing this instance of night and day, 
says : 

" It is necessary to our using the word cause, that we 
should believe not only that the antecedent always has 



194 MYSTICISM AND LOGIC 

been followed by the consequent, but that as long as the 
present constitution of things endures, it always will 
be so/ 1 

In this sense, we shall have to give up the hope of find 
ing causal laws such as Mill contemplated ; any causal 
sequence which we have observed may at any moment be 
falsified without a falsification of any laws of the kind 
that the more advanced sciences aim at establishing. 

In the fourth place, such laws of probable sequence, 
though useful in daily life and in the infancy of a science, 
tend to be displaced by quite different laws as soon as a 
science is successful. The law of gravitation will illustrate 
what occurs in any advanced science. In the motions of 
mutually gravitating bodies, there is nothing that can be 
called a cause, and nothing that can be called an effect ; 
there is merely a formula. Certain differential equations 
can be found, which hold at every instant for every 
particle of the system, and which, given the configuration 
and velocities at one instant, or the configurations at two 
instants, render the configuration at any other earlier or 
later instant theoretically calculable. That is to say, the 
configuration at any instant is a function of that instant 
and the configurations at two given instants. This state 
ment holds throughout physics, and not only in the special 
case of gravitation. But there is nothing that could be 
properly called " cause " and nothing that could be 
properly called " effect " in such a system. 

No doubt the reason why the old " law of causality " 
has so long continued to pervade the books of philo 
sophers is simply that the idea of a function is unfamiliar 
to most of them, and therefore they seek an unduly 
simplified statement. There is no question of repetitions 
of the " same " cause producing the " same " effect ; it 

Loc. cit.. 6- 



ON THE NOTION OF CAUSE 195 

is not in any sameness of causes and effects that the con 
stancy of scientific law consists, but in sameness of 
relations. And even " sameness of relations " is too 
simple a phrase ; " sameness of differential equations " 
is the only correct phrase. It is impossible to state this 
accurately in non-mathematical language ; the nearest 
approach would be as follows : " There is a constant 
relation between the state of the universe at any instant 
and the rate of change in the rate at which any part of 
the universe is changing at that instant, and this relation 
is many-one, i.e. such that the rate of change in the 
rate of change is determinate when the state of the 
universe is given." If the " law of causality " is to be 
something actually discoverable in the practice of science, 
the above proposition has a better right to the name 
than any " law of causality " to be found in the books of 
philosophers. 

In regard to the above principle, several observations 
must be made 

(1) No one can pretend that the above principle is a 
priori or self-evident or a " necessity of thought." Nor 
is it, in any sense, a premiss of science : it is an empirical 
generalisation from a number of laws which are them 
selves empirical generalisations. 

(2) The law makes no difference between past and 
future : the future " determines " the past in exactly 
the same sense in which the past " determines " the future. 
The word " determine," here, has a purely logical signifi 
cance : a certain number of variables " determine " 
another variable if that other variable is a function of 
them. 

(3) The law will not be empirically verifiable unless 
the course of events within some sufficiently small volume 



196 MYSTICISM AND LOGIC 

will be approximately the same in any two states of the 
universe which only differ in regard to what is at a con 
siderable distance from the small volume in question. 
For example, motions of planets in the solar system must 
be approximately the same however the fixed stars may 
be distributed, provided that all the fixed stars are very 
much farther from the sun than the planets are. If 
gravitation varied directly as the distance, so that the 
most remote stars made the most difference to the 
motions of the planets, the world might be just as regular 
and just as much subject to mathematical laws as it is at 
present, but we could never discover the fact. 

(4) Although the old " law of causality " is not assumed 
by science, something which we may call the " uniformity 
of nature " is assumed, or rather is accepted on inductive 
grounds. The uniformity of nature does not assert the 
trivial principle " same cause, same effect," but the 
principle of the permanence of laws. That is to say, 
when a law exhibiting, e.g. an acceleration as a function 
of the configuration has been found to hold throughout 
the observable past, it is expected that it will continue 
to hold in the future, or that, if it does not itself hold, 
there is some other law, agreeing with the supposed law 
as regards the past, which will hold for the future. The 
ground of this principle is simply the inductive ground 
that it has been found to be true in very many instances ; 
hence the principle cannot be considered certain, but 
only probable to a degree which cannot be accurately 
estimated. 

The uniformity of nature, in the above sense, although 
it is assumed in the practice of science, must not, in its 
generality, be regarded as a kind of major premiss, with 
out which all scientific reasoning would be in error. The 
assumption that all laws of nature are permanent has, of 



ON THE NOTION OF CAUSE 197 

course, less probability than the assumption that this or 
that particular law is permanent ; and the assumption 
that a particular law is permanent for all time has less 
probability than the assumption that it will be valid up 
to such and such a date. Science, in any given case, will 
assume what the case requires, but no more. In con 
structing the Nautical Almanac for 1915 it will assume 
that the law of gravitation will remain true up to the end 
of that year ; but it will make no assumption as to 1916 
until it comes to the next volume of the almanac. This 
procedure is, of course, dictated by the fact that the 
uniformity of nature is not known a priori, but is an 
empirical generalisation, like " all men are mortal." In 
all such cases, it is better to argue immediately from the 
given particular instances to the new instance, than to 
argue by way of a major premiss ; the conclusion is only 
probable in either case, but acquires a higher probability 
by the former method than by the latter. 

In all science we have to distinguish two sorts of laws : 
first, those that are empirically verifiable but probably 
only approximate ; secondly, those that are not verifiable, 
but may be exact. The law of gravitation, for example, 
in its applications to the solar system, is only empirically 
verifiable when it is assumed that matter outside the 
solar system may be ignored for such purposes ; we 
believe this to be only approximately true, but we cannot 
empirically verify the law of universal gravitation which 
we believe to be exact. This point is very important in 
connection with what we may call " relatively isolated 
systems." These may be defined as follows : 

A system relatively isolated during a given period is 
one which, within some assignable margin of error, will 
behave in the same way throughout that period, however 
the rest of the universe may be constituted. 



N 



198 MYSTICISM AND LOGIC 

A system may be called " practically isolated " during a 
given period if, although there might be states of the rest 
of the universe which would produce more than the 
assigned margin of error, there is reason to believe that 
such states do not in fact occur. 

Strictly speaking, we ought to specify the respect in 
which the system is relatively isolated. For example, 
the earth is relatively isolated as regards falling bodies, 
but not as regards tides ; it is practically isolated as 
regards economic phenomena, although, if Jevons sun- 
spot theory of commercial crises had been true, it would 
not have been even practically isolated in this respect. 

It will be observed that we cannot prove in advance 
that a system is isolated. This will be inferred from the 
observed fact that approximate uniformities can be 
stated for this system alone. If the complete laws for 
the whole universe were known, the isolation of a system 
could be deduced from them ; assuming, for example, 
the law of universal gravitation, the practical isolation of 
the solar system in this respect can be deduced by the 
help of the fact that there is very little matter in its 
neighbourhood. But it should be observed that isolated 
systems are only important as providing a possibility of 
discovering scientific laws ; they have no theoretical 
importance in the finished structure of a science. 

The case where one event A is said to " cause " another 
event B, which philosophers take as fundamental, is 
really only the most simplified instance of a practically 
isolated system. It may happen that, as a result of 
general scientific laws, whenever A occurs throughout a 
certain period, it is followed by B ; in that case, A and B 
form a system which is practically isolated throughout 
that period. It is, however, to be regarded as a piece of 
good fortune if this occurs ; it will always be due to special 



ON THE NOTION OF CAUSE 199 

circumstances, and would not have been true if the rest 
of the universe had been different though subject to the 
same laws. 

The essential function which causality has been sup 
posed to perform is the possibility of inferring the future 
from the past, or, more generally, events at any time from 
events at certain assigned times. Any system in which 
such inference is possible may be called a " determin 
istic " system. We may define a deterministic system as 
follows : 

A system is said to be " deterministic " when, given 
certain data, e v e 2 , . . ., e M , at times t lt tf a , . . ., t n respec 
tively, concerning this system, if E, is the state of the 
system at any time t, there is a functional relation of the 

form !,< ...,, <,*). (A) 



The system will be " deterministic throughout a given 
period " if t, in the above formula, may be any time 
within that period, though outside that period the 
formula may be no longer true. If the universe, as a 
whole, is such a system, determinism is true of the 
universe ; if not, not. A system which is part of a deter 
ministic system I shall call " determined " ; one which is 
not part of any such system I shall call " capricious." 

The events e lt e t , ___ , e n I shall call " determinants " 
of the system. It is to be observed that a system which 
has one set of determinants will in general have many. 
In the case of the motions of the planets, for example, 
the configurations of the solar system at any two given 
times will be determinants. 

We may take another illustration from the hypothesis 
of psycho-physical parallelism. Let us assume, for the 
purposes of this illustration, that to a griven state of brain 



200 MVSTICISM AND LOGIC 

a given state of mind always corresponds, and vice versa, 
i.e. that there is a one-one relation between them, so that 
each is a function of the other. We may also assume, 
what is practically certain, that to a given state of a 
certain brain a given state of the whole material universe 
corresponds, since it is highly improbable that a given 
brain is ever twice in exactly the same state. Hence 
there will be a one-one relation between the state of a 
given person s mind and the state of the whole material 
universe. It follows that, if n states of the material 
universe are determinants of the material universe, then 
n states of a given man s mind are determinants of the 
whole material and mental universe assuming, that is 
to say, that psycho-physical parallelism is true. 

The above illustration is important in connection with 
a certain confusion which seems to have beset those who 
have philosophised on the relation of mind and matter 
It is often thought that, if the state of the mind is deter 
minate when the state of the brain is given, and if the 
material world forms a deterministic system, then mind 
is " subject " to matter in some sense in which matter is 
not " subject " to mind. But if the state of the brain is 
also determinate when the state of the mind is given, it 
must be exactly as true to regard matter as subject to 
mind as it would be to regard mind as subject to matter. 
We could, theoretically, work out the history of mind 
without ever mentioning matter, and then, at the end, 
deduce that matter must meanwhile have gone through 
the corresponding history. It is true that if the relation 
of brain to mind were many-one, not one-one, there would 
be a one-sided dependence of mind on brain, while con 
versely, if the relation were one-many, as Bergson sup 
poses, there would be a one-aided dependence of brain on 
mind. But the dependence involved is. in any case, onlv 



ON THE NOTION OF CAUSE 201 

logical ; it does not mean that we shall be compelled to 
do things we desire not to do, which is what people in 
stinctively imagine it to mean. 

As another illustration we may take the case of 
mechanism and teleology. A system may be defined as 
" mechanical " when it has a set of determinants that 
are purely material, such as the positions of certain pieces 
of matter at certain times. It is an open question whether 
the world of mind and matter, as we know it, is a 
mechanical system or not ; let us suppose, for the sake 
of argument, that it is a mechanical system. This sup 
position so I contend throws no light whatever on the 
question whether the universe is or is not a " teleo- 
logical " system. It is difficult to define accurately what 
is meant by a " teleological " system, but the argument 
is not much affected by the particular definition we adopt. 
Broadly, a teleological system is one in which purposes 
are realised, i.e. in which certain desires those- that are 
deeper or nobler or more fundamental or more universal 
or what not are followed by their realisation. Now the 
fact if it be a fact that the universe is mechanical has 
no bearing whatever on the question whether it is teleo 
logical in the above sense. There might be a mechanical 
system in which all wishes were realised, and there might 
be one in which all wishes were thwarted. The question 
whether, or how far, our actual world is teleological, 
cannot, therefore, be settled by proving that it is mechani 
cal, and the desire that it should be teleological is no 
ground for wishing it to be not mechanical. 

There is, in all these questions, a very great difficulty 
in avoiding confusion between what we can infer and 
what is in fact determined. Let us consider, for a 
moment, the various senses in which the future may be 
" determined." There is one sense and a very important 



202 MYSTICISM AND LOGIC 

one in which it is determined quite independently of 
scientific laws, namely, the sense that it will be what it 
will be. We all regard the past as determined simply by 
the fact that it has happened ; but for the accident that 
memory works backward and not forward, we should 
regard the future as equally determined by the fact that 
it will happen. " But/ we are told, " you cannot alter 
the past, while you can to some extent alter the future." 
This view seems to me to rest upon just those errors in 
regard to causation which it has been my object to remove. 
You cannot make the past other than it was true, but 
this is a mere application of the law of contradiction. If 
you already know what the past was, obviously it is use 
less to wish it different. But also you cannot make the 
future other than it will be ; this again is an application 
of the law of contradiction. And if you happen to know 
the future e.g. in the case of a forthcoming eclipse it 
is just as useless to wish it different as to wish the past 
different. " But/ it will be rejoined, " our wishes can 
cause the future, sometimes, to be different from what it 
would be if they did not exist, and they can have no 
such effect upon the past." This, again, is a mere 
tautology. An effect being defined as something subse 
quent to its cause, obviously we can have no effect upon 
the past. But that does not mean that the past would 
not have been different if our present wishes had been 
different. Obviously, our present wishes are conditioned 
by the past, and therefore could not have been different 
unless the past had been different ; therefore, if our 
present wishes were different, the past would be different. 
Of course, the past cannot be different from what it was, 
but no more can our present wishes be different from what 
they are ; this again is merely the law of contradiction. 
The facts seem to be merely (i) that wishing generallv 



ON THE NOTION OF CAUSE 203 

depends upon ignorance, and is therefore commoner in 
regard to the future than in regard to the past ; (2) that 
where a wish concerns the future, it and its realisation 
very often form a " practically independent system," 
i.e. many wishes regarding the future are realised. But 
there seems no doubt that the main difference in our 
feelings arises from the accidental fact that the past 
but not the future can be known by memory. 

Although the sense of " determined " in which the 
future is determined by the mere fact that it will be what 
it will be is sufficient (at least so it seems to me) to refute 
some opponents of determinism, notably M. Bergson and 
the pragmatists, yet it is not what most people have in 
mind when they speak of the future as determined. What 
they have in mind is a formula by means of which the 
future can be exhibited, and at least theoretically calcu 
lated, as a function of the past. But at this point we 
meet with a great difficulty, which besets what has been 
said above about deterministic systems, as well as what 
is said by others. 

If formulae of any degree of complexity, however great, 
are admitted, it would seem that any system, whose 
state at a given moment is a function of certain measur 
able quantities, must be a deterministic system. Let us 
consider, in illustration, a single material particle, whose 
co-ordinates at time t are x t , y t , z t . Then, however, the 
particle moves, there must be, theoretically, functions 
/i /> /a, such that 

*,= /i(fl. *=/., *,=/ 

It follows that, theoretically, the whole state of the 
material universe at time / must be capable of being 
exhibited as a function of t. Hence our universe will be 
deterministic in the sense defined above. But if this be 



204 MYSTICISM AND LOGIC 

true, no information is conveyed about the universe in 
stating that it is deterministic. It is true that the formulae 
involved may be of strictly infinite complexity, and there 
fore not practically capable of being written down or 
apprehended. But except from the point of view of our 
knowledge, this might seem to be a detail : in itself, if 
the above considerations are sound, the material universe 
must be deterministic, must be subject to laws. 

This, however, is plainly not what was intended. The 
difference between this view and the view intended may 
be seen as follows. Given some formula which fits the 
facts hitherto say the law of gravitation there will be 
an infinite number of other formulae, not empirically dis 
tinguishable from it in the past, but diverging from it 
more and more in the future. Hence, even assuming 
that there are persistent laws, we shall have no reason 
for assuming that the law of the inverse square will hold 
in future ; it may be some other hitherto indistinguishable 
law that will hold. We cannot say that every law which 
has held hitherto must hold in the future, because past 
facts which obey one law will also obey others, hitherto 
indistinguishable but diverging in future. Hence there 
must, at every moment, be laws hitherto unbroken which 
are now broken for the first time. What science does, in 
fact, is to select the simplest formula that will fit the facts. 
But this, quite obviously, is merely a methodological 
precept, not a law of Nature. If the simplest formula 
ceases, after a time, to be applicable, the simplest formula 
that remains applicable is selected, and science has no 
sense that an axiom has been falsified. We are thus left 
with the brute fact that, in many departments of science, 
quite simple laws have hitherto been found to hold. This 
fact cannot be regarded as having any a priori ground, 
nor can it be used to support inductively the opinion that 



ON THE NOTION OF CAUSE 105 

the same laws will continue ; for at every moment laws 
hitherto true are being falsified, though in the advanced 
sciences these laws are less simple than those that have 
remained true. Moreover it would be fallacious to argue 
inductively from the state of the advanced sciences to the 
future state of the others, for it may well be that the 
advanced sciences are advanced simply because, hitherto, 
their subject-matter has obeyed simple and easily 
ascertainable laws, while the subject-matter of other 
sciences has not done so. 

The difficulty we have been considering seems to be 
met partly, if not wholly, by the principle that the time 
must not enter explicitly into our formulae. All mechanical 
laws exhibit acceleration as a function of configuration, 
not of configuration and time jointly ; and this principle 
of the irrelevance of the time may be extended to all 
scientific laws. In fact we might interpret the " uni 
formity of nature " as meaning just this, that no scientific 
law involves ttie time as an argument, unless, of course, 
it is given in an integrated form, in which case lapse of 
time, though not absolute time, may appear in our 
formulas. Whether this consideration suffices to over 
come our difficulty completely, I do not know ; but in 
any case it does much to diminish it. 

It will serve to illustrate what has been said if we apply 
it to the question of free will. 

(i) Determinism in regard to the will is the doctrine 
that our volitions belong to some deterministic system, 
i.e. are " determined " in the sense defined above. 
Whether this doctrine is true or false, is a mere question 
of fact ; no a priori considerations (if our previous dis 
cussions have been correct) can exist on either side. On 
the one hand, there is no a priori category of causality, 
but merely certain observed uniformities As a matter 



ao6 MYSTICISM AND LOGIC 

of fact, there are observed uniformities in regard to 
volitions ; thus there is some empirical evidence that 
volitions are determined. But it would be very rash to 
maintain that the evidence is overwhelming, and it is 
quite possible that some volitions, as well as some other 
things, are not determined, except in the sense in which 
we found that everything must be determined. 

(2) But, on the other hand, the subjective sense oi 
freedom, sometimes alleged against determinism, has no 
bearing on the question whatever. The view that it has 
a bearing rests upon the belief that causes compel their 
effects, or that nature enforces obedience to its laws as 
governments do. These are mere anthropomorphic 
superstitions, due to assimilation of causes with volitions 
and of natural laws with human edicts. We feel that our 
will is not compelled, but that only means that it is not 
other than we choose it to be. It is one of the demerits 
of the traditional theory of causality that it has created 
an artificial opposition between determinism and the 
freedom of which we are introspectively conscious. 

(3) Besides the general question whether volitions are 
determined, there is the further question whether they 
are mechanically determined, i.e. whether they are part 
of what was above defined as a mechanical system. This 
is the question whether they form part of a system with 
purely material determinants, i.e. whether there are laws 
which, given certain material data, make all volitions 
functions of those data. Here again, there is empirical 
evidence up to a point, but it is not conclusive in regard 
to all volitions. It is important to observe, however 
that even if volitions are part of a mechanical system, 
this by no means implies any supremacy of matter over 
mind. It mav well be that the same system which is 



ON THE NOTION OF CAUSE 207 

susceptible of material determinants is also susceptible 
of mental determinants ; thus a mechanical system may 
be determined by sets of volitions, as well as by sets of 
material facts. It would seem, therefore, that the reasons 
which make people dislike the view that volitions are 
mechanically determined are fallacious. 

(4) The notion of necessity, which is often associated 
with determinism, is a confused notion not legitimately 
deducible from determinism. Three meanings are 
commonly confounded when necessity is spoken of : 

(a) An action is necessary when it will be performed 
however much the agent may wish to do otherwise. 
Determinism does not imply that actions are necessary 
in this sense. 

(ft) A prepositional, function is necessary when att its 
values are true. This sense is not relevant to our present 
discussion. 

(y) A proposition is necessary with respect to a given 
constituent when it is the value, with that constituent as 
argument, of a necessary prepositional function, in other 
words, when it remains true however that constituent 
may be varied. In this sense, in a deterministic system, 
the connection of a volition with its determinants is 
necessary, if the time at which the determinants occur be 
taken as the constituent to be varied, the time-interval 
between the determinants and the volition being kept 
constant. But this sense of necessity is purely logical, 
and has no emotional importance. 

We may now sum up our discussion of causality. We 
found first that the law of causality, as usually stated by 
philosophers, is false, and is not employed in science. We 
then considered the nature of scientific laws, and found 
, instead of stating that one event A is always followed 



ao8 MYSTICISM AND LOGIC 

by another event B, they stated functional relations 
between certain events at certain times, which we called 
determinants, and other events at earlier or later times 
or at the same time. We were unable to find any a priori 
category involved : the existence of scientific laws ap 
peared as a purely empirical fact, not necessarily universal, 
except in a trivial and scientifically useless form. We 
found that a system with one set of determinants may very 
likely have other sets of a quite different kind, that, for 
example, a mechanically determined system may also be 
teleologically or volitionally determined. Finally we 
considered the problem of free will : here we found that 
the reasons for supposing volitions to be determined are 
strong but not conclusive, and we decided that even if 
volitions are mechanically determined, that is no reason 
for denying freedom in the sense revealed by intro 
spection, or for supposing that mechanical events are not 
determined by volitions. The problem of free will versus 
determinism is therefore, if we were right, mainly illusory, 
but in part not yet capable of being decisively solved. 



KNOWLEDGE BY ACQUAINTANCE 

AND KNOWLEDGE BY 

DESCRIPTION 

r I "HE object of the following paper is to consider what 
-* it is that we know in cases where we know pro 
positions about "the so-and-so " without knowing who 
or what the so-and-so is. For example, I know that the 
candidate who gets most votes will be elected, though I 
do not know who is the candidate who will get most 
votes. The problem I wish to consider is : Wha.^ fa we 
know in these cases, where the subject is merely described? 
I have considered this problem elsewhere 1 from a purely 
logical point of view ; bvit in what follows I Jjvish to con- 
sider the question in relation to theory of knowledge as 
well as in relation to logic, and in view of the above- 
mentioned logical discussions, I shall in this paper make 
the logical portion as brief as possible. 

In order to make clear the antithesis between " ac 
quaintance ""a^d " description/ I shall first of all try to 
explain what I mean by " acquaintance." I say that I 
am acquainted with an object when I have a direct 
cognitive relation to that object, i.e. when I am directly 
aware of the object itself. When I speak of a cognitive 
relation here, I^do not mean the sort of relation which 
constitutes judgment, but the sort which constitutes 
In fact, I think the relation of subject ano 

1 Soe references later. 
309 



aio MYSTICISM AND LOGIC 

object which I call acquaintance is simply the converse 
of the relation of object and subject which constitutes 
presentation. That is, to say that S has acquaintance 
with O is essentially the same thing as to say that is 
presented to S. But the associations and natural exten 
sions of the word acquaintance are different from those of 
the word presentation. To begin with, as in most cog 
nitive words, it is natural to say that I_am acquainted 
with an object even at moments when it is not actually 
before my mind, provided it has been before my mind, 
and will be again whenever occasion arises. This is the 
same sense in which I am said to know that 2+2=4 even 
when I am thinking of something else. In the second 

place, the word acquaintance is designed to emphasise, 

more than the word presentation, the relational character 
of the fact with which we are concerned. There is, to my 
mind, a danger that, in speaking of presentation, we 
may so emphasis the object as to lose sight of the sub 
ject. The result of this is either to lead to the view 
that there is no subject, whence we arrive at materialism ; 
or to lead to the view that what is presented is part of 
the subject, whence we arrive at idealism, and should 
arrive at solipsism but for the most desperate contortions. 
Now I wish to preserve the dualism of subject and object 
in^my terminology, because this dualism seems to me a 
fundamental fact concerning cognition. Hence I prefer 
the wordjacquaintance^ because it emphasises the need of 
a subject Avhich is acquainted. 

When /we ask what are the kinds of objects with which 
we are acquainted, the first and most obvious example is 
[sense-data. When I see a colour or hear a noise, I have 
direct acquaintance with the colour or the noise. The 
sense-datum with which I am acquainted in these cases 
is generally, if not always, complex. This is particularly 



KNOWLEDGE BY ACQUAINTANCE 211 

obvious in the case of sight. I do not mean, of course, 
merely that the supposed physical object is complex) but 
that the direct sensible obiecljs_jcornplex_and contains 
parts with spatial relations. Whether it is possible to be 
aware of a complex without being aware of its con 
stituents is not an easy question, but on the whole it 
would seem that there is no reason why it should not 
be possible. This question arises in an acute form in 
connection with self-consciousness, which we must now 
briefly consider. 

In introspection, we seem to be immediately aware of 
varying complexes, consisting of objects in various cog 
nitive and conative relations to ourselves. When I see 
the sun, it often happens that I am aware of my seeing 
the sun, in addition to being aware of the sun ; and when 
I desire food, it often happens that I am aware of my 
desire for food. /But it is hard to discover any state of 
mind in which I am aware of myself alone, as opposed to 
a complex of which I am a constituent./ Thp question oi 
the nature L of self-conscinnsnessjs too large, and too slightly 
connected with our subject, to be argued at length here. It 
is difficult, but probably not impossible, to account for 
plain facts if we assume that we do notjiave acquaintance 
with ourselves It is plain that we are not only acquainted 
with the complex " Self-acquainted-with-A," but we also 
know the proposition " I am acquainted with A." Now 
here the complex has been analysed, and if " I " does not 
stand for something which is a direct object of acquaint 
ance, we shall have to syppose that " I is something 
kno wr^ by description , af we wished to maintain the view 
thai. there is nn acquaiutajic^jwith Self, we might argue 
a,s... follows ; We are acquainted with acquaintance, and 
we know that it is a relation. Also we are acquainted 
with a corn pi ex in which we perceive that acquaintance 

wQ Afi. iJ/Vi u bUNn* -j f\ ;wU>A 






212 MYSTICISM AND LOGIC 

isjhe relating relation . Hence we know that this complex 
must have a constituent which is that which is acquainted, 
i.e. must have a subject-term as well as an object-term. 
This subject-term we define as " I." Thus " I " means 
" the subject-term in awarenesses of which 7 am aware." 
But as a definition this cannot be regarded as a happy 
effort. It would seem necessary, therefore, either to 
suppose that I am acquainted with myself, and that " I," 
therefore, requires no definition, being merely the proper 
name of a certain object, or to find some other analysis 
of self -consciousness. Thus self -consciousness cannot be 
regarded as throwing light on the question whether we 
(Jan know a complex without knowing its constituents?) 
This question, however, is not important for our present 
purposes, and I shall therefore not discuss it further. 

The awarenesses we have considered so far have all 
been awarenesses of particular^ existents, and might all 
in a large sense be called sense-data. For, from the point 
of view ojQheofyjtf knowledge, introspective knowledge 
is exactly on a level with knowledge derived from sight 
or hearing. But, in addition to awareness of the above 
kind of objects, which may be called awareness 
of particulars), we have also (though not quite in 
the same sense) what may be called awarenessoL 
universals. Awareness jofjuniversals is called conceiving,^ 
and a universal of which we are aware is called a concept. 
Not only are we aware of particular yellows, but if we 
\ have seen a sufficient number of yellows and have suffi- 
~~ cient intelligence, we are aware of the universal yellow ;. , 
w this universal is the subject in such judgments as " yellow 
differs from blue " or " yellow resembles blue less than 
green does." And the universal yellow is the predicate in 
such judgments as " this is yellow," where " this " is a 
particular sense-datum. And universal relations, too. 



KNOWLEDGE BY ACQUAINTANCE 213 

are objects of awarenesses ; up and down, before and 
after, resemblance, desire, awareness itself, and so on, 
would seem to be all of them objects of which we can be 
aware, w f ..j*urJw c i 

In regard to relations, it might be urged that we are 
never aware of the- universal relation/itself , but only of 
complexes in which it is a constituent. For example, it 
may be said that we do not know directly such a relation 
as before, though we understand such a proposition as 
" this is before that/ and may be directly aware of such 
a complex as " this being before that." This view, how 
ever, is difficult to reconcile with the fact that we often 
know propositions in which the relation is the subject, 
or in which the relata are not definite given objects, but 
" anything." For example, we know that if one thing is 
before another, and the other before a third, then the 
first is before the third ; and here the things concerned 
are not definite things, but "anything." It is hard to 
see how we could know such a fact about " before " 
unless we were acquainted with " before," and not merely 
with actual particular cases of one given object being 
before another given object. And more directly : A 
judgment such as " this is before that," where this judg 
ment is derived from awareness of a complex, constitutes 
an analysis, and we should not understand the analysis if 
we were not acquainted with the meaning of the terms 
employed. Thus we must suppose that we are acquainted 
with the meaning of " before," and not merely with 
instances of it. v - 

There are thus at least, two sorts of objects of which we 
are aware, namely, particulars and universals. Among 
particulars I include alljsxistents, and all complexes of 
which one_orjmore constituents are existents, such as 
this^before-that, this-above-that, the-yellowness-of-this. 



114 MYSTICISM AND LOGIC 

Among universals I jnchide all objects of which no par- 
IP ticular is a constituent. Thus the disjunction " universal- 
particular " includes all objects. We might also call it the 
disjunction " abstract-concrete." It is not quite parallel 
with the opposition " concept-percept," because things 
remembered or imagined belong with particulars, but can 
hardly be called percepts. (On the other hand, universals 
with which we are acquainted may be identified with 
concepts.)JU^ 

It__will be seen that among the objects with which we 

are acquainted are not included physical objects (as 

opposed to sense-data), nortrther people s minds. These 

things are known to us by what I call " knowledge by 

/^description fl^hich we must now consider. 

By a " description " I mean any phrase of the form "a 
so-and-so " or " the so-and-so." A phrase of the form 
^j^feo-and^sp " I shall call an " ambiguous " description ; 

a phrase of the form " the so-and-so " (in the singular) I 

^jO 

shall call a "definite " description, yfih us "a man " is 
an ambiguous description, and " the man with the iron 
mask " is a definite description./ There are various 
problems connected with ambiguous descriptions, but I 
pass them by, since they do not directly concern the matter 
I wish to discuss. What I wish tojiiscuss is the nature of 
our knowledge concerning objects in cases where we know 
that there is an object answering to a definite description, 

with any such object. This 



is a matter which is concerned exclusively with definite 
descriptions. I shall, therefore, in the sequel, speak 
simply of " descriptions " when I mean " definite descrip 
tions." Thus a description will mean any phrase of the 
form ^jthe so-and-so " in the singular, v 

I shall say that an object is " known by description " 
when we know that it is " the so-and-so," i.e. when we 



KNOWLEDGE BY ACQUAINTANCE 

know that there is one object, and no more, having a 
certain property ; and it_ will generally be implied that 
we do not have knowledge of the same object by ac 
quaintance. We know that the man with the iron mask 
existed, and many propositions are known about him ; 
but we do not know who he was. We know that the 
candidate who gets most votes will be elected, and in this 
case we are very likely also acquainted (in the only sense 
in which one can be acquainted with some one else) with 
the man who is, in fact, the candidate who will get most 
votes, but we do not know which of the candidates he is, 
i.e. we do not know any proposition of the form " A is 
the candidate who will get most votes " where A is one 
of the candidates by name. We shall say that we have 
"^merely descriptive knowledge/ of the so-and-so when, 
although we know that the so-and-so exists/and although 
we may possibly be acquainted with the object which is, 
in fact, the so-and-so, yet we do not know any pro- 
position " a is the so-and-so," where a is something with 
which we are acquainted. ^ 

When we say " the^ so-and-so ^xists," we mean that 
there is just one object which is the so-and-so. The pro 
position "a is the so-and-so" means that a has the 
property so-and-so, and nothing else has. " Sir Joseph 
Larmor is the Unionist candidate " means " Sir Joseph 
Larmor is a Unionist candidate, and no one else is." 
" The Unionist candidate exists " means " some one is a 
Unionist candidate, and no one else is." Thus, when we 
are acquainted with an object which we know to be the 
so-and-so, we know that the so-and-so exists) but we_may 
know that the so-and-so exists when we are not acquainted 
with any object which we know to be the so-and-so, and 
even when we are not acquainted with any object which, in 
tact, is the so-and-so. 



ai6 MYSTICISM AND LOGIC 

Common words, even proper names, a-re_usuall^_realiy 
descriptions. Jfhat is to say, the thought in the mind of 
a person using a proper name correctly can generally only 
be expressed explicitly- if we replace the proper name by 
a description. Moreover, the description required to 
express the thought will vary for different people, or for 
the same person at different times. The only thing 
constant (so long as the name is rightly used) is _the object 
to which the name applies. But so long as this remains 
constant, the particular description involved usually 
makes no difference to the truth or falsehood of the pro 
position in which the name appears. 

Let us take some illustrations. Suppose some state 
ment made about Bismarck. Assuming that there is 
such a thing as direct acquaintance with oneself, Bismarck 
himself might have used his name directly to designate 
the particular person with whom he was acquainted. In 
this case, if he made a judgment about himself, he him 
self might be a constituent of the judgment. Here the 
proper name has the direct use which it always wishes to 
have, as simply standing for a certain object, and not 
for a description of the object. But if a person who knew 
Bismarck made a judgment about him, the case is 
different. What this person was acquainted with were 
certain sense-data which he connected (rightly, we will 
suppose) with Bismarck s body. jpHis body as a physical 
object, and still more his mind, were only known as the 
body and the mind connected with these sense-data. 
That is, they were known by descriptionj&It is, of course, 
very much a matter of chance which characteristics of a 
man s appearance will come into a friend s mind when 
he thinks of him ; thus the description actually in the 
friend s mind is accidental;!^ The essential point is that 
he knows that the various descriptions all apply to the 



KNOWLEDGE BY ACQUAINTANCE 217 

same entity, in spite of not being acquainted with the 
entity in question. 

When we, who did not know Bismarck, make a judg 
ment about him, the description in our minds will probably 
be some more or less vague mass of historical knowledge 
far more, in most cases, than is required to identify 
him. But, for the sake of illustration, let us assume that 
we think of him as " the first Chancellor of the German 
Empire." Here all the words are abstract! except " Ger 
man/ The word " German " will again have different 
meanings for different people. To some it will recall 
travels in Germany, to some the look of Germany on the 
map, and so on. But if we are to obtain a description 
which we know to be applicable, we shall be compelled, 
at some point, to bring in a reference to a particular with 
which_we _a_re_ acquainted. v Such reference is involved in 
any mention of past, present, and future (as opposed to 
definite dates), orj)fjiere jind there, or of what others 
have told us. Thus it would seem that, in some way or 
other, a description known to be applicable to a particular 
must involve some reference to a particular with which 
we are acquainted, if our knowledge about the thing 
described is not to be merely what follows logically from 
the description. For example, " the most long-lived of 
men is a description which must apply to some man, 
but we can make no judgments concerning this man 
which involve knowledge about him beyond what the 
description gives. ; If, however, we say, " the first 
Chancellor of the German Empire was an astute diplo 
matist," we can only be assured of the truth of our 
judgment in virtue of something with which we are 
acquainted usually a testimony heard or read. Con 
sidered psychologically, apart from the information we 
convey to others, apart from the fact about the actual 



*i8 MYSTICISM AND LOGIC 

Bismarck, which gives importance to our judgment, the 
thought we reall^haye contains the one or more par 
ticulars involved, iq.d otherwise consists wholly of con 
cepts. All names of places- London, England, Europe, 
the earth, the Solar System similarly involve, when 
used, descriptions which start from some one or more 
particulars with which we are acquainted. I suspect that 
even the Universe, as considered by metaphysics, involves 
such a connection with particulars. In logic, on the 
contrary, where we are concerned not merely with what 
does exist, but with whatever might or could exist or be, 
no reference to actual particulars is involved.^^JcCcY 
It would seem that, when we make a statement about 
something only known by description, we often intend to 
make our statement, not in the form involving the 
description, but about the actual thing described/ That 
is to say, when we say anything about Bismarck, we 
should like, if we could, to make the judgment which 
Bismarck alone can make, namely, the judgment of which 
he himself is a constituent. In this we are necessarily 
defeated, since the actual Bismarck is unknown to us. 
But we know that there is an object B called Bismarck, 
and that B was an astute diplomatist. We can thus 
describe the proposition we should like to affirm, namely, 
" B was an astute diplomatist," where B is the object 
which was Bismarck. WhaL_ejiables us to communicate 
in spite of the varying descriptions we employ is that we 
know there is a true proposition concerning the actual 
Bismarck, and that, however we may vary the description 
(so_long as the description is correct), the proposition 

described is still the same. This proposition, which is 
described and is known to be true, is what interests us ; 

| but we are not acquainted with the proposition itself, 
and do not know it, though we know it is true./ 



KNOWLEDGE BY ACQUAINTANCE 219 

It will be seen that there are various stages in the 
removal from acquaintance with particulars : there is 
Bismarck to people who knew him/Bismarck to those 
who only know of him through history K the man with the 
iron mask, the longest-lived of men^-^Tfiese are progres 
sively further removed from acquaintance with particulars, 
and there is a similar hierarchy in the region of universals. 
Many universals, like many particulars, are only known k 
to us by description^ T But here, as in the case of particu 
lars, knowledge concerning what is known by description 
is~ultlmately reducible to knowledge concerning what is 
knownBy"ac^aint a nc e . \S 

The^Tundamental epistemological principle in the 
analysis of propositions containing descriptions is this : 
Every proposition which we can understand must be com- 
posed wholly of constituents with which we are acquainted. :- 
FronTwhat has been said already, it will be plain why I 
advocate this principle, and how I propose to meet the 
case of propositions which at first sight contravene it. 
Let us begin with the reasons for supposing the principle 
true. 

The chief reason for supposing the principle true is 
that it seems scarcely possible to believe that we can 
make a judgment or entertain a supposition without 
knowing what it is that we are judging or supposing 
about. If we make a judgment about (say) Julius Caesar, 
it is plain that the actual person who was Julius Caesar is 
not a constituent of the judgment. But before going 
further, it may be well to explain what I mean when I 
say that this or that is a constituent of a judgment, or of 
a proposition which we understand. To begin with 
judgments : a judgment, as an occurrence, I take to be 
a relation of a mind to several entities, namely, the 
entities which compose what is judgM. If, e.g. I judge 



320 MYSTICISM AND LOGIC 

that A ioves B, the judgment as an event consists in the 
existence, at a certain moment, of a specific four-term 
relation, called judging, between me and A and love and 
B. That is to say, at the time when I judge, there is a 
certain complex whose terms are myself and A and love 
ind B, and whose relating relation is judging. My reasons 
for this view have been set forth elsewhere, 1 and I shall not 
repeat them here. Assuming this view of judgment, the 
constituents of the judgment are simply the constituents of 
the complex which is the judgment. Thus, in the above 
case, the constituents are myself and A and love and B 
and judging. But myself and judging are constituents 
shared by all my judgments ; thus the distinctive con- 
stituents of the particular judgment in question are A 
and love and B. Coming now to what is meant by 
" understanding a proposition," I should say that there 
is another relation possible between me and A and love 
and B, which is called my (upposing that A loves B. 2 
When we can suppose that A loves B, we " understand 
the proposition " Amoves B. ffihus we often understand a 
proposition in cases where we have not enough knowledge 
Supposing, like judging, is a many- 



term relation, of which a mind is one Term. The other 
terms of the relation are called the constituents of the 
proposition supposed. Thus the principle which I 
enunciated may be re-stated as follows \KWhenever a 

1 Philosophical Essays, " The Nature of Truth." I have been per 
suaded by Mr. Wittgenstein that this theory is somewhat unduly 
simple, but the modification which I believe it to require does not 
affect the_ab.Qve argument [1917]. 

* Cf. Meinong, Ueber Annahmen, passim. I formerly supposed, 
contrary to Meinong s view, that the relationship of supposing might 
be merely that of presentation. In this view I now think I was mis 
taken, and Meinong is right. But my present view depends upon the 
theory that both in judgment and in assumption there is no single 
Objective, but the several constituents of the judgment or assumption 
are in a many-term relation to the mind. 



KNOWLEDGE BY ACQUAINTANCE 221 

relation of supposing or judging occurs, the terms to which 
the supposing or judging mind is related by the relation of 
supposing or judging must be terms with which the mind in 
question is acquainted. This is merely to say that we 
cannot make a judgment or a supposition without know 
ing what it is that we are making our judgment or sup 
position about. It seems to me that the truth of this 
principle is evident as soon as the principle is understood ; 
I shall, therefore, in what follows, assume the principle, 
and use it as a guide in analysing judgments that contain 
descriptions. / 

Returning now to Julius Caesar, I assume that it will 
be admitted that he himself is not a constituent of any 
judgment which I can make. But at this point it is 
necessary to examine the view that judgments are com 
posed of something called " ideas," and that it is the 
" idea " of Julius Caesar that is a constituent of my 
judgment. I believe the plausibility of this view rests 
upon a failure to form a right theory of descriptions. We 
may mean by my " idea " of Julius Caesar the things that 
I know about him, e.g. that he conquered Gaul, was 
assassinated on the Ides of March, and is a plague to 
schoolboys. Now I am admitting, and indeed contending, 
that in order to discover what is actually in my mind 
when I judge about Julius Caesar, we must substitute for 
the proper name a description made up of some of the 
thing? I know about him. (A description which will 
often serve to express my thought is " the man whose 
name was Julius Ccesar." For whatever else I may have 
forgotten about him, it is plain that when I mention him 
I have not forgotten that that was his name.) But 
although I think the theory that judgments consist of 
ideas may have been suggested in some such way, yet I 
think the theory itself is fundamentally mistaken. The 



222 MYSTICISM AND LOGIC 

view seems to be that there is some mental existent 
which may be called the " idea " of something outside 
the mind of the person who has the idea, and that, since 
judgment is a mental event, its constituents must be 
constituents of the mind of the person judging. But in 
this view ideas become a veil between us and outside 
things we never really, in knowledge, attain to the 
things^w,e are supposed to be knowing about, but only to 
thq(ideasx)f those things.} The relation of mind, idea, and 
object, on this view, is utterly obscure, and, so far as I 
can see, nothing discoverable by inspection warrants the 
intrusion of the idea between the mind and the object. 
I suspect that the view is fostered by the dislike of 
relations7)and that it is felt the mind could not know 
objects unless there were something "in " the mind 
which could be called the state of knowing the object. 
Such a view, however, leads at once to a vicious endless 
regress, since the relation of idea to object will have to be 
explained by supposing that the idea itself has an idea of 
the object, and so on ad infinitum. I therefore see no 
reason to believe that, when we are acquainted with an 
object, there is in us something which can be called the 
" idea " of the object, ^^he^^ntrajy,^! hold that 
acquaintance is wholly a relation, not demanding any 
such constituent of the mind as is supposed by advocates 
of " ideas." This is, ofcourse, a large question, and one 
wnich would take us far from our subject if it were 
adequately discussed. I therefore content myself with 
the above indications, arid with the corollary that, in 
judging, the actual objects concerning which we judge, 
rather than any supposed purely mental entities, are 
constituents of the complex which is the judgment. 

When, therefore, I say that we must substitute for 
"Julius Caesar" some description of Julius Caesar, in order 



/ 
KNOWLEDGE BY ACQUAINTANCE 223 

to discover the meaning of a judgment nominally about 
him, I am not saying that we must substitute an idea. 
Suppose our description is " the man whose name was 
Julius Ccesar" Let our judgment be " Julius Caesar was 
assassinated." Then it becomes Vjhe man whose name 
was Julius Ccesar was assassinated."] Here Julius Casar 
is a noise or shape with which we are acquainted, and all 
the other constituents of the judgment (neglecting the 
tense in "was") are concepts with which we are_ ac 
quainted. Thus ou judgment is wholly reduced to_ con 
stituents with which we are acquainted, but Julius Caesar 
himself has ceased to be a constituent of our judgment. 
This, however, requires a proviso, to be further explained 
shortly, namely _ that 1* the man whose name was Julius 
Ccesar " must not, as a whole, be a constituent of our 
judgment, that is to say, this phrase must not, as a whole, 
have a meaning which enters into the judgment. Any 
right analysis of the judgment, therefore, must break up 
this phrase, and not treat it as a subordinate complex 
which is part of the judgment. The judgment " the man 
whose name was Julius C<zar was assassinated " may 
be interpreted as meaning f" one and only one man was, 
called Julius Casar, and that one was assassinated."; 
Here it is plain that there is no constituent corresponding 
to the phrase " the man whose name was Julius Cczsar." 
Thus there is no reason to regard this phrase as expressing 
a constituent of the judgment, and we have seen that this 
phrase must be broken up if we are to be acquainted with 
all the constituents of the judgment. This conclusion, 
which we have reached from considerations concerned 
with the theory of knowledge, is also forced upon us by 
logical considerations, which must now be briefly re 
viewed. 

It is common to distinguish two aspects, meaning ahd 



224 MYSTICISM AND LOGIC 

denotation, in such phrases as " the author of Waverley." 
The meaning will be a certain complex, consisting (at 
least) of authorship and Waverley (with some relation/; 
denotation will be Scott./ Similarly "featherless 
bipeds " will have a complex meajiing, containing as 
constituents the presence of two feet and the absence of 
feathers,^) while its denotation will be the class of men. 
Thus when we say " Scott is the author of Waverley " or 
" men are the same as featherless bipeds," we are assert 
ing an identity of denotation, and this assertion is wortr 
making because of the diversity, of meaning. 1 I believe 
that the duality of meaning and denotation, though 
capable of a true interpretation, is misleading if taken as 
fundamental. The denotation, I believe, is not a con 
stituent of the proposition, except in the case of proper 
names, i.e. of words which do not assign a property to 
an object, but merely and solely name it. And I should 
hold further that, in this sense, there are only two words. 
which ^re strictly proper names of particulars, namely, 



One reason for not believing the denotation to be a con 
stituent of the proposition is that we may know the pro 
position even when we are not acquainted with the 
denotation. The proposition "the author of Waverley 
is a novelist " was known to people who did not know 
that " the author of Waverley " denoted Scott. This 
reason has been already sufficiently emphasised. 

A second reason is that propositions concerning " the 
so-and-so " are possible even^vhen "the so-and-so " has 
no denotation. Take, e.g. [ the golden mountain doei 
not exist "/orr the round square is self-contradictory J3 

1 This view has been recently advocated by Miss E. E. C. Jones. 
" A New Law of Thought and its Implications," Mind, January, 1911, 
I should now exclude " I " from proper names in the strict sense. 
and retain only "this" 1017]. 



KNOWLEDGE BY ACQUAINTANCE 225 

If we are to preserve the duality of meaning a^id denota 
tion, we have to say, with Meinong, that there are such 
objects as the golden mountain and the round square, 
although these objects do not have being. We even have 
to admit that the existent round square is existent, but 
does not exist. 1 Meinong does not regard this as a con 
tradiction, but I fail to see that it is noJ,Dne. Indeed, it 
seems to me evident that the judgment t there is no such 
object as the round square"; does not presuppose that 
there is such an object. If this is admitted, however, we 
are led to the conclusion that, by parity of form, no judg 
ment concerning " the so-and-so " actually involves the 
so-and-so as a constituent. 

Miss Jones 2 contends that there is no difficulty in admit 
ting^, contradictory predicates concerning such an object 
as (Jthe present King of France, ! on the ground that this 
object is in itself contradictory. ""Now it might, of course, 
be argued that this object, unlike the round square, is 
not self -contradictory, but merely non-existent. This, 
however, would not go to the root of the matter. The 
real objection to such an argument is that the law of 
contradiction ought not to be stated in the traditional 
form " A is not both B and not B," but in the form " nc 
proposition is both true and false." The traditional form 
only applies to certain propositions, namely, to those 
which attribute a predicate to a subject. When the law 
is stated of propositions, instead of being stated concern 
ing subjects and predicates, it is at once evident that 
propositions about the present King of France or the 
round square can form no exception, but are just as in 
capable of being both true and false as other propositions. 
Miss Jones 8 argues that " Scott is the author of 

1 Meinong, Utber Annahmen, 2nd ed., Leipzig, 1910, p. 141. 
1 Mind, July, 1910, p. 380. * Mind, July, 1910, p. 379. 



226 MYSTICISM AND LOGIC 

Waverley " asserts identity of denotation between Scott 
and the author of Waverley. But there is some difficulty 
in choosing among alternative meanings of this con 
tention. In the first place, it should be observed that the 
author of Waverley is not a mere name, like Scott. Scott is 
merely a noise or shape conventionally used to designate 
a certain person ; it gives us no information about that 
person, and has nothing that can be called meaning as 
opposed to denotation. (I neglect the fact, considered 
above, that even proper names, as a rule, really stand for 
descriptions.) But the author of Waverley is not merely 
conventionally a name for Scott ; the element of mere 
convention belongs here to the separate words, the and 
author and of and Waverley. Given what these words 
stand for, the author of Waverley is no longer arbitrary. 
When it is said that Scott is the author of Waverley, we 
are not stating that these are two names for one man, as 
we should be if we said " Scott is Sir Walter." A man s 
name is what he is called, but however much Scott had 
been called the author of Waverley, that would not have 
made him be the author ; it was necessary for him 
actually to write Waverley, which was a fact having 
nothing to do with names. 

If. then, we are asserting identity of denotation, we 
must not mean by denotation the mere relation of a name 
to the thing named. In fact, it would be nearer to the 
truth to say that the meaning of " Scott " is the denota 
tion of " the author of Waverley." The relation oi 
" Scott " to Scott is that " Scott " means Scott, just as 
the relation of " author " to the concept which is so called 
is that " author " means this concept. Thus if we 
distinguish meaning and denotation in " the author of 
Waverley," we shall have to say that " Scott " has mean 
ing but not denotation. Also when we say " Scott is the 



KNOWLEDGE BY ACQUAINTANCE 227 

author of Waverley," the meaning of " the author of 
Waverley " is relevant to our assertion. For if the 
denotation alone were relevant, any other phrase with 
the same denotation would give the same proposition. 
Thus " Scott is the author of Marmion " would be the 
same proposition as " Scott is the author of Waverley/ 
But this is plainly not the case, since from the first we 
learn that Scott wrote Marmion and from the second we 
learn that he wrote Waverley, but the first tells us 
nothing about Waverley and the second nothing about 
Marmion. Hence the meaning of " the author of Waver 
ley/ as opposed to the denotation, is certainly relevant 
to " Scott is the author of Waverley." 

We have thus agreed that " the author of Waverley " 
is not a mere name, and that its meaning is relevant in 
propositions in which it occurs. Thus if we are to say, as 
Miss Jones does, that " Scott is the author of Waverley " 
asserts an identity of denotation, we must regard the 
denotation of " the author of Waverley " as the denota 
tion of what is meant by " the author of Waverley." Let 
us call the meaning of " the author of Waverley " M. 
Thus M is what " the author of Waverley " means. Then 
we are to suppose that " Scott is the author of Waverley " 
means " Scott is the denotation of M." But here we are 
explaining our proposition by another of the same form, 
and thus we have made no progress towards a real 
explanation. " The denotation of M," like " the author 
of Waverley," has both meaning and denotation, on the 
theory we are examining If we call its meaning M , our 
proposition becomes " Scott is the denotation of M ." 
But this leads at once to an endless regress. Thus the 
attempt to regard our proposition as asserting identity 
of denotation breaks down, and it becomes imperative 
to find some other analysis. When this analysis has been 



228 MYSTICISM AND LOGIC 

completed, we shall be able to reinterpret the phrase 
" identity of denotation," which remains obscure so long 
as it is taken as fundamental. 

The first point to observe is that, in any proposition 
about " the author of Waverley," provided Scott is not 
explicitly mentioned, the denotation itself, i.e. Scott, 
does not occur, but only the concept of denotation, which 
will be represented by a variable. Suppose we say " the 
author of Waverley was the author of Marmion," we are 
certainly not saying that both were Scott we may have 
forgotten that there was such a person as Scott. We are 
saying that there is some man who was the author of 
Waverley and the author of Marmion. That is to say, 
there is some one who wrote Waverley and Marmion, 
and no one else wrote them. Thus the identity is that 
of a variable, i.e. of an indefinite subject, " some one." 
This is why we can understand propositions about " the 
author of Waverley," without knowing who he was. 
When we say " the author of Waverley was a poet," we 
mean " one and only one man wrote Waverley, and he 
was a poet " ; when we say " the author of Waverley 
was Scott " we mean " one and only one man wrote 
Waverley, and he was Scott." Here the identity is 
between a variable, i.e. an indeterminate subject (" he "), 
and Scott ; " the author of Waverley " has been analysed 
away, and no longer appears as a constituent of the 
proposition. 1 

The reason why it is imperative to analyse away the 
phrase " the author of Waverley " may be stated as 
follows. It is plain that when we say " the author of 
Waverley is the author of Marmion," the is expresses 

1 The theory which I am advocating is set forth fully, with the> 
logical grounds in its favour, in Pnncipia Mathematical, Vol. I, Intro 
duction, Chap. ITT ; also, less fully, in Mind. October, 1905 



KNOWLEDGE BY ACQUAINTANCE 229 

identity. We have seen also that the common denotation. 
namely Scott, is not a constituent of this proposition, 
while the meanings (if any) of " the author of Waverley " 
and " the author of Marmion " are not identical. We 
have seen also that, in any sense in which the meaning of 
a word is a constituent of a proposition in whose verbal 
expression the word occurs, " Scott " means the 
actual man Scott, in the same sense (so far as concerns 
our present discussion) in which " author " means 
a certain universal. Thus, if " the author of Waverley " 
were a subordinate complex in the above proposition, its 
meaning would have to be what was said to be identical 
with the meaning of " the author of Marmion." This is 
plainly not the case ; and the only escape is to say that 
" the author of Waverley " does not, by itself, have a 
meaning, though phrases of which it is part do have a 
meaning. That is, in a right analysis of the above pro 
position, " the author of Waverley " must disappear. 
This is effected when the above proposition is analysed 
as meaning : " Some one wrote Waverley and no one 
else did, and that some one also wrote Marmion and no 
one else did." This may be more simply expressed by 
saying that the prepositional function " x wrote Waverley 
and Marmion, and no one else did " is capable of truth, 
i.e. some value of x makes it true, but no other value 
does. Thus the true subject of our judgment is a 
prepositional function, i.e. a complex containing an 
undetermined constituent, and becoming a proposition as 
soon as this constituent is determined. 

We may now define the denotation of a phrase. If we 
know that the proposition " a is the so-and-so " is true, 
i.e. that a is so-and-so and nothing else is, we call a the 
denotation of the phrase " the so-and-so." A very great 
many of the propositions we naturally make about " the 
p 



230 MYSTICISM AND LOGIC 

so-and-so " will remain true or remain false if we sub 
stitute a for " the so-and-so," where a is the denotation 
of " the so-and-so." Such propositions will also remain 
true or remain false if we substitute for " the so-and-so " 
any other phrase having the same denotation. Hence, 
as practical men, we become interested in the denotation 
more than in the description, since the denotation decides 
as to the truth or falsehood of so many statements in 
which the description occurs. Moreover, as we saw 
earlier in considering the relations of description and 
acquaintance, we often wish to reach the denotation, and 
are only hindered by lack of acquaintance : in such cases 
the description is merely the means we employ to get as 
near as possible to the denotation. Hence it naturally 
comes to be supposed that the denotation is part of the 
proposition in which the description occurs. But we 
have seen, both on logical and on epistemological grounds, 
that this is an error. The actual object (if any) which is 
the denotation is not (unless it is explicitly mentioned) a 
constituent of propositions in which descriptions occur ; 
and this is the reason why, in order to understand such 
propositions, we need acquaintance with the constituents 
of the description, but do not need acquaintance with its 
denotation. The first result of analysis, when applied to 
propositions whose grammatical subject is " the so-and- 
so," is to substitute a variable as subject ; i.e. we obtain 
a proposition of the form : " There is something which 
alone is so-and-so, and that something is such-and-such." 
The further analysis of propositions concerning " the so- 
and-so " is thus merged in the problem of the nature of 
the variable, i e. of the meanings of some, any, and all. 
This is a difficult problem, concerning which I do not 
intend to say anything at present. 

To sum up our whole discussion We began by dis- 



KNOWLEDGE BY ACQUAINTANCE 231 

tinguishing two sorts of knowledge of objects, namely, 
knowledge by acquaintance and knowledge by description. 
Of these it is only the former that brings the object itself 
before the mind. We have acquaintance with sense-data, 
with many universals, and possibly with ourselves, but 
not with physical objects or other minds. We have 
descriptive knowledge of an object when we know that it 
is the object having some property or properties with 
which we are acquainted ; that is to say, when we know 
that the property or properties in question belong to one 
object and no more, we are said to hav3 knowledge of 
that one object by description, whether or not we are 
acquainted with the object. Our knowledge of physical 
objects and of other minds is only knowledge by descrip 
tion, the descriptions involved being usually such as 
involve sense-data. All propositions intelligible to us, 
whether or not they primarily concern things only known 
to us by description, are composed wholly of constituents 
with which we are acquainted, for a constituent with which 
we are not acquainted is unintelligible to us. A judgment, 
we found, is not composed of mental constituents called 
" ideas," but consists of an occurrence whose con 
stituents are a mind 1 and certain objects, particulars 
or universals. (One at least must be a universal.) When 
a judgment is rightly analysed, the objects which are con 
stituents of it must all be objects with which the mind 
which is a constituent of it is acquainted. This con 
clusion forces us to analyse descriptive phrases occurring 
in propositions, and to say that the objects denoted by 
such phrases are not constituents of judgments in which 
such phrases occur (unless these objects are explicitly 

1 I use this phrase merely to denote the something psychological 
which enters into judgment, without intending to prejudge the 
question as to what this something is. 



232 MYSTICISM AND LOGIC 

mentioned). This leads us to the view (recommended 
also on purely logical grounds) that when we say " the 
author of Marmion was the author of Waverley," Scott 
himself is not a constituent of our judgment, and that 
the judgment cannot be explained by saying that it 
affirms identity of denotation with diversity of meaning. 
It also, plainly, does not assert identity of meaning. 
Such judgments, therefore, can only be analysed by 
breaking up the descriptive phrases, introducing a vari 
able, and making prepositional functions the ultimate 
subjects. In fact, " the so-and-so is such-and-such " will 
mean that " x is so-and-so and nothing else is, and x is 
such-and-such " is capable of truth. The analysis of 
such judgments involves many fresh problems, but the 
discussion of these problems is not undertaken in the 
present paper. 



INDEX 



Achilles and the tortoise, 80 ff, 

89 ff 

Acquaintance, the relation of, 209 ff 
Alexander, 125 
American Realists, the, 134 
Aristotle, 42, 76, 97 

Bacon, 41 

Bergson, 14 ff, 22, 105, 128, 185 ff, 

203 

Berkeley, 97, 132 
Blake, i 
Bosanquet, 99 
Broad, 89 n 

Calculus, the, 82 

Cantor, Georg, 64, 81 ff, 85, 91 

Carlyle, 50, 82 

Cause, the conception of, 135 , 

iSoff 

Christianity and renunciation, 51 
Chuang Tzu, 106 
Construction of permanent things and 

matter, 169 ff 
Constructions, logical, 155 ff 

Darwin, 15, 23. 43 
Dedekind, 64, 81 ff, 85 
Descartes, 97, 126 
Descriptions, 175, 214 tf 

Education, 37 ff 
Euclid, 62, 92, 94 
Evolutionism, 23 ft, 28 

Fano, 93 
Faraday, 34 
Free will, 205 ff 
Frege, 78 



Galileo, 42 
Gladstone, 177 
Good and evil, 26 ff 

Hegel, 8, 10, 18, 85, 97, 105 ff 

Heine, 113 

Heraclitus, I ff, IO 

Hertz, 34 

Holt, 177 n 

Hume, i, 97 

Infinite, the mathematical, 84 ff 

James, William, 100 

Jones, Miss E. E. C., 224 n, 22$ 

Judgment, 219 ff 

Kant, 85, 96, 97, 99, n8ff 
Knowledge by acquaintance, 209 ff ; 
by description, 214 ff 

Laplace, 23 

Leibniz, 76, 79, 82 ff, 97, 126, 144, 

160 

Locke, 97 
Logic, the laws of, 68 ff 

Macaulay and Taylor * theorem, 95 

Malthus, 43 

Mathematics, 58 ff; and the Meta 
physicians, 74 ff ; and logic, 75 ff : 
and the infinitesimal. 82 ff 

Matter, the nature of, 125 ff; defi 
nition of, 164 ff 

Maxwell, 34 

Meaning and denotation, 223 ff 

Meinong, 174, 220 , 225 

Militarism, 50 

Mill, 185, 193 ff 

Mysticism and logic, I ft 



*34 



MYSTICISM AND LOGIC 



Necessity, the notion of, 207 ff 
Nietzsche, 22, 50 
Nunn, 125, 137 , 153 

Parmenides, 7 ff, 18, 21 
Particulars, awareness of, 210 ff 
Peano, 78 ff, 93 ff 
Perspectives, 139 ff; the space of, 

158 ff 

Philosophy and logic, 1 1 1 
Physics, sense-data and, 145 ff 
Pierce, 76 n 

Plato, i ff, 10, 30, 60, 97 
Pragmatism, 22, 105 

Realism and the analytic method, 

120 ff 

Reason and intuition, 12 ff 
Relatives, the logic of, 76 
Robb, 167 n 

Santayana, 2O 

Sense-data, 147, 210 ff; and nhysics, 



Sensibilia, 148 ff 

Space, 138 ff; private, 158 ff; the 
logical problem, 114 ff; the prob 
lem in physics, 115 ff; the episte 
mological problem, 118 ff 

Systems, deterministic, 199 ; prac 
tically iiolated, 198 ; relatively 
isolated, 197 ; mechanical, 201 

Time, 10, 21 ff, 141 ff, 167 ff 
Tristram Shandy, the paradox of, 
90 ff 

Unity and Plurality, 18 ff 
Universals, awareness of, 212 ff 

Ward, 180 

Weierstrass, 80, 82, 95 
Whitehead, 117, 157, 175 
Wolf, 173 

Zeno the Eleatic, 64, 80, 84, 
89 ff 




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